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Answer test results

This page exposes the results of running answer tests on STACK test cases. This page is automatically generated from the STACK unit tests and is designed to show question authors what answer tests actually do. This includes cases where answer tests currentl fail, which gives a negative expected mark. Comments and further test cases are very welcome.

Answer test
Passed?
Student response
Teacher answer
Options
CAS errors
Raw mark
Expected mark
Feedback
Answer note
AlgEquiv Expected failure 1/0 1 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATAlgEquiv_STACKERROR_SAns.
AlgEquiv Expected failure 1 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATAlgEquiv_STACKERROR_TAns.
AlgEquiv Expected failure (x-1)^2 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question.
ATAlgEquivTEST_FAILED-Empty SA.
AlgEquiv Expected failure x^2 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty teacher answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty teacher answer, probably a CAS validation problem when authoring the question.
ATAlgEquivTEST_FAILED-Empty TA.
AlgEquiv Expected failure x-1)^2 (x-1)^2 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question.
ATAlgEquivTEST_FAILED-Empty SA.
AlgEquiv Pass x1 x_1 0 0
AlgEquiv Pass x_1 x[1] 0 0
AlgEquiv Pass x[1] x1 0 0
Predicates
AlgEquiv Pass integerp(3) true 1 1 ATLogic_True.
AlgEquiv Pass integerp(3.1) true 0 0
AlgEquiv Pass integerp(3) false 0 0
AlgEquiv Pass integerp(3) true 1 1 ATLogic_True.
AlgEquiv Pass lowesttermsp(x^2/x) true 1 1 ATLogic_True.
AlgEquiv Pass lowesttermsp(-y/-x) true 1 1 ATLogic_True.
AlgEquiv Pass lowesttermsp((x^2-1)/(x-1)) true 0 0
AlgEquiv Pass lowesttermsp((x^2-1)/(x+2)) true 1 1 ATLogic_True.
Case sensitivity
AlgEquiv Pass X x 0 0 ATAlgEquiv_WrongCase.
AlgEquiv Pass exdowncase(X) x 1 1
AlgEquiv Pass exdowncase((X-1)^2) x^2-2*x+1 1 1
Permutations of variables (To do: a dedicated answer test with feedback)
AlgEquiv Pass Y=1+X y=1+x 0 0 ATEquation_default
AlgEquiv Pass v+w+x+y+z a+b+c+A+B 0 0
Numbers
AlgEquiv Pass 4^(-1/2) 1/2 1 1
AlgEquiv Pass 4^(1/2) sqrt(4) 1 1
Mix of floats and rational numbers
AlgEquiv Pass 0.5 1/2 1 1
AlgEquiv Pass 0.33 1/3 0 0
AlgEquiv Pass 5.1e-2 51/1000 1 1
AlgEquiv Pass 0.333333333333333 1/3 0 0
AlgEquiv Pass (0.5+x)*2 2*x+1 1 1
Complex numbers
AlgEquiv Pass sqrt(-1) %i 1 1
AlgEquiv Pass %i e^(i*pi/2) 1 1
AlgEquiv Pass (4*sqrt(3)*%i+4)^(1/5) 8^(1/5)*(cos(%pi/15)+%i*sin(%pi/15)) 1 1
AlgEquiv Pass (4*sqrt(3)*%i+4)^(1/5) rectform((4*sqrt(3)*%i+4)^(1/5)) 1 1
AlgEquiv Pass (4*sqrt(3)*%i+4)^(1/5) polarform((4*sqrt(3)*%i+4)^(1/5)) 1 1
AlgEquiv Pass %i/sqrt(x) sqrt(-1/x) 1 1
Infinity
AlgEquiv Pass inf inf 1 1
AlgEquiv Pass inf -inf 0 0
AlgEquiv Pass 2*inf inf 0 0
AlgEquiv Pass 0*inf 0 1 1
Powers and roots
AlgEquiv Pass x^(1/2) sqrt(x) 1 1
AlgEquiv Pass x sqrt(x^2) 0 0
AlgEquiv Pass abs(x) sqrt(x^2) 1 1
AlgEquiv Pass 1/abs(x)^(1/3) (abs(x)^(1/3)/abs(x))^(1/2) 1 1
AlgEquiv Pass sqrt((x-3)*(x-5)) sqrt(x-3)*sqrt(x-5) 0 0
AlgEquiv Pass 1/sqrt(x) sqrt(1/x) 1 1
AlgEquiv Pass x-1 (x^2-1)/(x+1) 1 1
AlgEquiv Pass 2^((1/5.1)*t) 2^((1/5.1)*t) 1 1
AlgEquiv Pass 2^((1/5.1)*t) 2^(0.196078431373*t) 0 0
AlgEquiv Pass a^b*a^c a^(b+c) 1 1
AlgEquiv Pass (a^b)^c a^(b*c) 0 0
AlgEquiv Pass (assume(a>0),(a^b)^c) a^(b*c) 1 1
AlgEquiv Pass (assume(x>2),6*((x-2)^2)^k) 6*(x-2)^(2*k) 1 1
AlgEquiv Pass signum(-3) -1 1 1
AlgEquiv Pass 6*((x-2)^3)^k 6*(x-2)^(3*k) 1 1
AlgEquiv Pass (4*sqrt(3)*%i+4)^(1/5) 6^(1/5)*cos(%pi/15)-6^(1/5)*%i*sin(%pi/15) 0 0
AlgEquiv Pass 2+2*sqrt(3+x) 2+sqrt(12+4*x) 1 1
AlgEquiv Pass 6*e^(6*(y^2+x^2))+72*x^2*e^(6*(y^2+x^2)) (72*x^2+6)*e^(6*(y^2+x^2)) 1 1
Expressions with subscripts
AlgEquiv Pass a1 a_1 0 0
AlgEquiv Pass rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) 1 1
AlgEquiv Pass rho*z*V/(4*pi*epsilon[1]*(R^2+z^2)^(3/2)) rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) 0 0
AlgEquiv Pass sqrt(k/m)*sqrt(m/k) 1 1 1
AlgEquiv Pass (2*pi)/(k/m)^(1/2) (2*pi)/(k/m)^(1/2) 1 1
AlgEquiv Pass (2*pi)*(m/k)^(1/2) (2*pi)/(k/m)^(1/2) 1 1
AlgEquiv Pass sqrt(2*x/10+1) sqrt((2*x+10)/10) 1 1
AlgEquiv Pass ((x+3)^2*(x+3))^(1/3) ((x+3)*(x^2+6*x+9))^(1/3) 1 1
Need to factor internally.
AlgEquiv Pass ((x+3)^2*(x+3))^(1/3) ((x+3)*(x^2+6*x+9))^(1/3) 1 1
Polynomials and rational function
AlgEquiv Pass (x-1)^2 x^2-2*x+1 1 1
AlgEquiv Pass (x-1)*(x^2+x+1) x^3-1 1 1
AlgEquiv Pass (x-1)^(-2) 1/(x^2-2*x+1) 1 1
AlgEquiv Pass 1/(4*x-(%pi+sqrt(2))) 1/(x+1) 0 0
AlgEquiv Pass (x-a)^6000 (x-a)^6000 1 1
AlgEquiv Pass (a-x)^6000 (x-a)^6000 1 1
AlgEquiv Pass (4*a-x)^6000 (x-4*a)^6000 1 1
AlgEquiv Pass (x-a)^6000 (x-a)^5999 0 0
AlgEquiv Pass (k+8)/(k^2+4*k-12) (k+8)/(k^2+4*k-12) 1 1
AlgEquiv Pass (k+7)/(k^2+4*k-12) (k+8)/(k^2+4*k-12) 0 0
AlgEquiv Pass -(2*k+6)/(k^2+4*k-12) -(2*k+6)/(k^2+4*k-12) 1 1
AlgEquiv Pass 1/n-1/(n+1) 1/(n*(n+1)) 1 1
AlgEquiv Pass 0.5*x^2+3*x-1 x^2/2+3*x-1 1 1
AlgEquiv Pass 14336000000*x^13+250265600000*x^12+1862860800000*x^11+7623925760000*x^10+18290677760000*x^9+24744757985280*x^8+14567212351488*x^7-3267871272960*x^6-6408053107200*x^5+670406720000*x^4+1179708800000*x^3-429244800000*x^2+56696000000*x-2680000000 512*(2*x+5)^7*(5*x-1)^5*(70*x+67) 1 1
AlgEquiv Pass 14336000000*x^13+250265600000*x^12+1862860800000*x^11+7623925760000*x^10+18290677760000*x^9+24744757985280*x^8+14567212351488*x^7-3267871272960*x^6-6408053107200*x^5+670406720000*x^4+1179708800000*x^3-429244800000*x^2+56696000000*x-2680000001 512*(2*x+5)^7*(5*x-1)^5*(70*x+67) 0 0
AlgEquiv Pass 14336000000*x^13 512*(2*x+5)^7*(5*x-1)^5*(70*x+67) 0 0
Trig functions
AlgEquiv Pass cos(x) cos(-x) 1 1
AlgEquiv Pass cos(x)^2+sin(x)^2 1 1 1
AlgEquiv Pass cos(x+y) cos(x)*cos(y)-sin(x)*sin(y) 1 1
AlgEquiv Pass cos(x+y) cos(x)*cos(y)+sin(x)*sin(y) 0 0
AlgEquiv Pass cos(x#pm#y) cos(x)*cos(y)-(#pm#sin(x)*sin(y)) 1 1 ATLogic_True.
AlgEquiv Pass sin(x#pm#y) sin(x)*cos(y)#pm#cos(x)*sin(y) 1 1 ATLogic_True.
AlgEquiv Pass sin(x#pm#y) cos(x)*sin(y)#pm#sin(x)*cos(y) 0 0
AlgEquiv Pass 2*cos(x)^2-1 cos(2*x) 1 1
AlgEquiv Pass 1.0*cos(1200*%pi*x) cos(1200*%pi*x) 1 1
AlgEquiv Pass diff(tan(10*x)^2,x) cos(6*x) 0 0
AlgEquiv Pass exp(%i*%pi) -1 1 1
AlgEquiv Pass 2*cos(2*x)+x+1 -sin(x)^2+3*cos(x)^2+x 1 1
AlgEquiv Pass (2*sec(2*t)^2-2)/2 -(sin(4*t)^2-2*sin(4*t)+cos(4*t)^2-1)*(sin(4*t)^2+2*sin(4*t)+cos(4*t)^2-1)/(sin(4*t)^2+cos(4*t)^2+2*cos(4*t)+1)^2 1 1
AlgEquiv Pass 1+cosec(3*x) 1+csc(3*x) 1 1
AlgEquiv Pass 1/(1+exp(-2*x)) tanh(x)/2+1/2 1 1
AlgEquiv Pass 1+cosech(3*x) 1+csch(3*x) 1 1
AlgEquiv Pass -4*sec(4*z)^2*sin(6*z)-6*tan(4*z)*cos(6*z) -4*sec(4*z)^2*sin(6*z)-6*tan(4*z)*cos(6*z) 1 1
AlgEquiv Pass -4*sec(4*z)^2*sin(6*z)-6*tan(4*z)*cos(6*z) 4*sec(4*z)^2*sin(6*z)+6*tan(4*z)*cos(6*z) 0 0
AlgEquiv Pass csc(6*x)^2*(7*sin(6*x)*cos(7*x)-6*cos(6*x)*sin(7*x)) -(6*cos(6*x)*sin(7*x)-7*sin(6*x)*cos(7*x))/sin(6*x)^2 1 1
AlgEquiv Pass csc(6*x)^2*(7*sin(6*x)*cos(7*x)-6*cos(6*x)*sin(7*x)) (6*cos(6*x)*sin(7*x)-7*sin(6*x)*cos(7*x))/sin(6*x)^2 0 0
AlgEquiv Pass -(7*x^6+4*x^3)/sin(7*y+x^7+x^4+1)^2 -(7*x^6+4*x^3)*csc(7*y+x^7+x^4+1)^2 1 1
AlgEquiv Pass sin((2*%pi*n-%pi)/2) -cos(n*%pi) 1 1
AlgEquiv Pass sin(x/2)/(1+tan(x)*tan(x/2)) sin(x/2)*cos(x) 1 1
AlgEquiv Pass (declare(n,integer),trigrat(sin((2*%pi*n-%pi)/2))) -(-1)^n 1 1
AlgEquiv Pass cot(%pi/20)+cot(%pi/24)-cot(%pi/10) sqrt(1)+sqrt(2)+sqrt(3)+sqrt(4)+sqrt(5)+sqrt(6) 0 -3
AlgEquiv Pass trigeval(cot(%pi/20)+cot(%pi/24)-cot(%pi/10)) sqrt(1)+sqrt(2)+sqrt(3)+sqrt(4)+sqrt(5)+sqrt(6) 1 1
AlgEquiv Pass sin([1/8,1/6, 1/4, 1/3, 1/2, 1]*%pi) [sqrt(2-sqrt(2))/2,1/2,1/sqrt(2),sqrt(3)/2,1,0] 0 -3
The entries underlined in red below are those that are incorrect. \[\left[ {\color{red}{\underline{\sin \left( \frac{\pi}{8} \right)}}} , \frac{1}{2} , \frac{1}{\sqrt{2}} , \frac{\sqrt{3}}{2} , 1 , 0 \right] \]
(ATList_wrongentries 1).
AlgEquiv Pass trigeval(sin([1/8,1/6, 1/4, 1/3, 1/2, 1]*%pi)) [sqrt(2-sqrt(2))/2,1/2,1/sqrt(2),sqrt(3)/2,1,0] 1 1
Logarithms
AlgEquiv Pass log(a^2*b) 2*log(a)+log(b) 1 1
AlgEquiv Pass (2*log(2*x)+x)/(2*x) (log(2*x)+2)/(2*sqrt(x)) 0 0
AlgEquiv Pass log(abs((x^2-9))) log(abs(x-3))+log(abs(x+3)) 0 0
AlgEquiv Pass lg(10^x) x 1 1
AlgEquiv Pass lg(3^x,3) x 1 1
AlgEquiv Pass lg(a^x,a) x 1 1
AlgEquiv Pass 1+lg(27,3) 4 1 1
AlgEquiv Pass 1+lg(27,3) 3 0 0
AlgEquiv Pass lg(1/8,2) -3 1 1
AlgEquiv Pass lg(root(x,n)) lg(x,10)/n 1 1
AlgEquiv Pass log(root(x,n)) lg(x,10)/n 0 0
AlgEquiv Pass x^log(y) y^log(x) 1 1
Hyperbolic trig
AlgEquiv Pass e^1-e^(-1) 2*sinh(1) 1 1
Lists
AlgEquiv Pass x [1,2,3] 0 0
Your answer should be a list, but is not. Note that the syntax to enter a list is to enclose the comma separated values with square brackets.
ATAlgEquiv_SA_not_list.
AlgEquiv Pass [1,2] [1,2,3] 0 0
Your list should have \(3\) elements, but it actually has \(2\).
ATList_wronglen.
AlgEquiv Pass [1,2,4] [1,2,3] 0 0
The entries underlined in red below are those that are incorrect. \[\left[ 1 , 2 , {\color{red}{\underline{4}}} \right] \]
(ATList_wrongentries 3).
AlgEquiv Pass [1,x>2] [1,2<x] 1 1
AlgEquiv Pass [1,2,[2-x<0,{1,2,2,2, 1,3}]] [1,2,[2-x<0,{1,2}]] 0 0
The entries underlined in red below are those that are incorrect. \[\left[ 1 , 2 , \left[ 2-x < 0 , \left \{1 , 2 , 3 \right \} \right] \right] \]
(ATList_wrongentries 3: (ATList_wrongentries 2: ATSet_wrongsz)).
AlgEquiv Pass [(k+8)/(k^2+4*k-12),-(2*k+6)/(k^2+4*k-12)] [(k+8)/(k^2+4*k-12),-(2*k+6)/(k^2+4*k-12)] 1 1
Rounding of floats
AlgEquiv Pass round(0.5) 0.0 1 1
AlgEquiv Pass round(1.5) 2.0 1 1
AlgEquiv Pass round(2.5) 2.0 1 1
AlgEquiv Pass round(12.5) 12.0 1 1
AlgEquiv Pass significantfigures(0.5,1) 0.5 1 1
AlgEquiv Pass significantfigures(1.5,1) 2.0 1 1
AlgEquiv Pass significantfigures(2.5,1) 3.0 1 1
AlgEquiv Pass significantfigures(3.5,1) 4.0 1 1
AlgEquiv Pass significantfigures(11.5,2) 12.0 1 1
AlgEquiv Pass 1500 scientific_notation(1500,3) 1 1
AlgEquiv Pass 1500 displaysci(1.5,2,3) 1 1
AlgEquiv Pass [3,3.1,3.14,3.142,3.1416,3.14159,3.141593,3.1415927] makelist(significantfigures(%pi,i),i,8) 1 1
Sets
AlgEquiv Pass x {1,2,3} 0 0
Your answer should be a set, but is not. Note that the syntax to enter a set is to enclose the comma separated values with curly brackets.
ATAlgEquiv_SA_not_set.
AlgEquiv Pass co(1,2) {1,2,3} 0 0
Your answer should be a set, but is not. Note that the syntax to enter a set is to enclose the comma separated values with curly brackets.
ATAlgEquiv_SA_not_set.
AlgEquiv Pass {1,2} {1,2,3} 0 0
Your set should have \(3\) different elements, but it actually has \(2\).
ATSet_wrongsz.
AlgEquiv Pass {2/4, 1/3} {1/2, 1/3} 1 1
AlgEquiv Pass {A[1],A[2],A[4]} {A[1],A[2],A[3]} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{A_{4} \right \}\]
ATSet_wrongentries.
AlgEquiv Pass {A[1],A[2],A[3]} {A[1],A[2],A[3]} 1 1
AlgEquiv Pass {1,2,4} {1,2,3} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{4 \right \}\]
ATSet_wrongentries.
AlgEquiv Pass {1,x>4} {4<x, 1} 1 1
AlgEquiv Pass {x-1=0,x>1 and 5>x} {x>1 and x<5,x=1} 1 1
AlgEquiv Pass {x-1=0,x>1 and 5>x} {x>1 and x<5,x=2} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{x-1=0 \right \}\]
ATSet_wrongentries.
AlgEquiv Pass {x-1=0,x>1 and 5>x} {x>1 and x<3,x=1} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{5-x > 0\,{\mbox{ and }}\, x-1 > 0 \right \}\]
ATSet_wrongentries.
Equivalence for elements of sets is different from expressions: see docs.
AlgEquiv Pass {-sqrt(2)/sqrt(3)} {-2/sqrt(6)} 0 -3
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{-\frac{\sqrt{2}}{\sqrt{3}} \right \}\]
ATSet_wrongentries.
AlgEquiv Pass {[-sqrt(2)/sqrt(3),0],[2/sqrt(6),0]} {[2/sqrt(6),0],[-2/sqrt(6),0]} 0 -3
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{\left[ -\frac{\sqrt{2}}{\sqrt{3}} , 0 \right] \right \}\]
ATSet_wrongentries.
AlgEquiv Pass ev(radcan({-sqrt(2)/sqrt(3)}),simp) ev(radcan({-2/sqrt(6)}),simp) 1 1
AlgEquiv Pass ev(radcan(ratsimp({(-sqrt(10)/2)-2,sqrt(10)/2-2},algebraic:true)),simp) ev(radcan(ratsimp({(-sqrt(5)/sqrt(2))-2,sqrt(5)/sqrt(2)-2},algebraic:true)),simp) 1 1
AlgEquiv Pass {(2-2^(5/2))/2,(2^(5/2)+2)/2} {1-2^(3/2),2^(3/2)+1} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{\frac{2-2^{\frac{5}{2}}}{2} , \frac{2^{\frac{5}{2}}+2}{2} \right \}\]
ATSet_wrongentries.
AlgEquiv Pass ev(radcan({(2-2^(5/2))/2,(2^(5/2)+2)/2}),simp) {1-2^(3/2),2^(3/2)+1} 1 1
AlgEquiv Pass {(x-a)^6000} {(a-x)^6000} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{{\left(x-a\right)}^{6000} \right \}\]
ATSet_wrongentries.
AlgEquiv Pass {(k+8)/(k^2+4*k-12),-(2*k+6)/(k^2+4*k-12)} {(k+8)/(k^2+4*k-12),-(2*k+6)/(k^2+4*k-12)} 1 1
Matrices
AlgEquiv Pass matrix([1,2],[2,3]) matrix([1,2],[2,3]) 1 1
AlgEquiv Pass matrix([1,2],[2,3]) matrix([1,2,3],[2,3,3]) 0 0
Your matrix should be \(2\) by \(3\), but it is actually \(2\) by \(2\).
ATMatrix_wrongsz_columns.
AlgEquiv Pass matrix([1,2],[2,3]) matrix([1,2],[2,5]) 0 0
The entries underlined in red below are those that are incorrect. \[ \left[\begin{array}{cc} 1 & 2 \\ 2 & {\color{red}{\underline{3}}} \end{array}\right]\]
ATMatrix_wrongentries.
AlgEquiv Pass matrix([0.33,1],[1,1]) matrix([0.333,1],[1,1]) 0 0
The entries underlined in red below are those that are incorrect. \[ \left[\begin{array}{cc} {\color{red}{\underline{0.33}}} & 1 \\ 1 & 1 \end{array}\right]\]
ATMatrix_wrongentries.
AlgEquiv Pass matrix([x+x,2],[2,x*x]) matrix([2*x,2],[2,x^2]) 1 1
AlgEquiv Pass matrix([epsilon[0],2],[2,x^2]) matrix([epsilon[0],2],[2,x^2]) 1 1
AlgEquiv Pass matrix([epsilon[2],2],[2,x^2]) matrix([epsilon[0],2],[2,x^3]) 0 0
The entries underlined in red below are those that are incorrect. \[ \left[\begin{array}{cc} {\color{red}{\underline{\varepsilon_{2}}}} & 2 \\ 2 & {\color{red}{\underline{x^2}}} \end{array}\right]\]
ATMatrix_wrongentries.
AlgEquiv Pass matrix([x>4,{1,x^2}],[[1,2],[1,3]]) matrix([4-x<0,{x^2, 1}],[[1,2],[1,3]]) 1 1
AlgEquiv Pass matrix([x>4,{1,x^2}],[[1,2],[1,3]]) matrix([4-x<0,{x^2, 1}],[[1,2],[1,4]]) 0 0
The entries underlined in red below are those that are incorrect. \[ \left[\begin{array}{cc} x > 4 & \left \{1 , x^2 \right \} \\ \left[ 1 , 2 \right] & \left[ 1 , {\color{red}{\underline{3}}} \right] \end{array}\right]\]
ATMatrix_wrongentries.
Vectors
AlgEquiv Pass a stackvector(a) 0 0
Equations
AlgEquiv Pass 1 x=1 0 0
Your answer should be an equation, but is not.
ATAlgEquiv_SA_not_equation.
AlgEquiv Pass x=1 x=1 1 1 ATEquation_sides
AlgEquiv Pass 1=x 1=x 1 1 ATEquation_sides
AlgEquiv Pass 1=x x=1 1 1 ATEquation_sides_op
AlgEquiv Pass 1=1 1=x 0 0 ATEquation_default
AlgEquiv Pass 1=1 x=1 0 0 ATEquation_default
AlgEquiv Pass x=2 x=1 0 0 ATEquation_lhs_notrhs
AlgEquiv Pass 2=x x=1 0 0 ATEquation_default
AlgEquiv Pass x=x y=y 1 1 ATEquation_zero
AlgEquiv Pass x+y=1 y=1-x 1 1
AlgEquiv Pass 2*x+2*y=1 y=0.5-x 1 1 ATEquation_ratio
AlgEquiv Pass 1/x+1/y=2 y = x/(2*x-1) 1 1 ATEquation_ratio
AlgEquiv Pass y=sin(2*x) y/2=cos(x)*sin(x) 1 1 ATEquation_ratio
AlgEquiv Pass y=(x-a)^6000 y=(x-a)^6000 1 1 ATEquation_sides
AlgEquiv Pass y=(x-a)^5999 y=(x-a)^6000 0 0 ATEquation_lhs_notrhs
AlgEquiv Pass y=(a-x)^6000 y=(x-a)^6000 1 1 ATEquation_sides
AlgEquiv Pass y=(a-x)^5999 y=(x-a)^5999 0 0 ATEquation_lhs_notrhs
AlgEquiv Pass y=(a-x)^59999 y=(x-a)^5999 0 0 ATEquation_lhs_notrhs
AlgEquiv Pass x+y=i y=i-x 1 1
AlgEquiv Pass (1+%i)*(x+y)=0 y=-x 1 1
AlgEquiv Pass s^2*%e^(s*t)=0 s^2=0 0 0 ATEquation_default
AlgEquiv Pass x^6000-x^6001=x^5999 x^5999*(1-x+x^2)=0 1 1 ATEquation_ratio
AlgEquiv Pass x^6000-x^6001=x^5999 x^5999*(1-x+x^3)=0 0 0 ATEquation_default
AlgEquiv Pass 258552*x^7*(81*x^8+1)^398 x^3*(x^4+1)^399 0 0
AlgEquiv Pass Ia*(R1+R2+R3)-Ib*R3=0 -Ia*(R1+R2+R3)+Ib*R3=0 1 1
AlgEquiv Pass a=0 or b=0 a*b=0 1 1 ATEquation_sides
AlgEquiv Pass a*b=0 a=0 or b=0 1 1 ATEquation_sides
AlgEquiv Pass a*x=a*y x=y 0 0 ATEquation_default
AlgEquiv Pass a*x=a*y a=0 or x=y 1 1 ATEquation_ratio
Unary Equations
AlgEquiv Pass 1 stackeq(1) 1 1
AlgEquiv Pass stackeq(1) 1 1 1
AlgEquiv Pass stackeq(1) 0 0 0
Equations: Loose/gain roots with nth powers of each side.
AlgEquiv Pass x=y x^2=y^2 0 0 ATEquation_default
AlgEquiv Pass (x-2)^2=0 x=2 0 0 ATEquation_default
AlgEquiv Pass a^3*b^3=0 a=0 or b=0 0 0 ATEquation_default
AlgEquiv Pass a^3*b^3=0 a*b=0 0 0 ATEquation_default
AlgEquiv Pass (x-y)*(x+y)=0 x^2=y^2 1 1 ATEquation_ratio
AlgEquiv Pass x=1 (x-1)^3=0 0 0 ATEquation_default
AlgEquiv Pass sqrt(x)=sqrt(y) x=y 0 0 ATEquation_default
AlgEquiv Pass x=sqrt(a) x^2=a 0 0 ATEquation_default
AlgEquiv Pass (x-sqrt(a))*(x+sqrt(a))=0 x^2=a 1 1 ATEquation_ratio
AlgEquiv Pass (x-%i*sqrt(a))*(x+%i*sqrt(a))=0 x^2=-a 1 1 ATEquation_ratio
AlgEquiv Pass (x-%i*sqrt(abs(a)))*(x+%i*sqrt(abs(a)))=0 x^2=-abs(a) 1 1 ATEquation_ratio
AlgEquiv Pass y=sqrt(1-x^2) x^2+y^2=1 0 0 ATEquation_default
AlgEquiv Pass (y-sqrt(1-x^2))*(y+sqrt(1-x^2))=0 x^2+y^2=1 1 1 ATEquation_ratio
AlgEquiv Pass (y-sqrt((1-x)*(1+x)))*(y+sqrt((1-x)*(1+x)))=0 x^2+y^2=1 1 1 ATEquation_ratio
AlgEquiv Pass (x-1)*(x+1)*(y-1)*(y+1)=0 y^2+x^2=1+x^2*y^2 1 1 ATEquation_ratio
Equations: edge cases. Teacher must enter an equation, all or none here.
AlgEquiv Pass all x=x 1 1 ATEquation_zero
AlgEquiv Pass true x=x 1 1 ATEquation_zero
AlgEquiv Pass x=x all 1 1 ATEquation_zero
AlgEquiv Pass all all 1 1 ATEquation_zero
AlgEquiv Pass true all 1 1 ATEquation_zero
AlgEquiv Pass a=a x=x 1 1 ATEquation_zero
AlgEquiv Pass false x=x 0 0 ATEquation_zero_fail
AlgEquiv Pass false all 0 0 ATEquation_zero_fail
AlgEquiv Pass none all 0 0 ATEquation_zero_fail
AlgEquiv Pass all none 0 0 ATEquation_empty_fail
AlgEquiv Pass 2=3 1=4 1 1 ATEquation_empty
AlgEquiv Pass none 1=2 1 1 ATEquation_empty
AlgEquiv Pass false 1=2 1 1 ATEquation_empty
AlgEquiv Pass none none 1 1 ATEquation_empty
AlgEquiv Pass false none 1 1 ATEquation_empty
AlgEquiv Pass 3=0 none 1 1 ATEquation_empty
AlgEquiv Pass 0=3 none 1 1 ATEquation_empty
AlgEquiv Pass all 1=2 0 0 ATEquation_empty_fail
AlgEquiv Pass true 1=2 0 0 ATEquation_empty_fail
AlgEquiv Pass {} 1=2 0 0
Your answer should be an equation, but is not.
ATAlgEquiv_SA_not_equation.
AlgEquiv Pass [] 1=2 0 0
Your answer should be an equation, but is not.
ATAlgEquiv_SA_not_equation.
AlgEquiv Pass {} none 0 0
Your answer should be an equation, inequality or a logical combination of many of these, but is not.
ATAlgEquiv_SA_not_logic.
Sets of real numbers
AlgEquiv Pass x^2 cc(1,3) 0 0
Your answer should be a subset of the real numbers. This could be a set of numbers, or a collection of intervals.
ATAlgEquiv_SA_not_realset.
AlgEquiv Pass %union(oo(1,2),oo(3,4)) %union(oo(1,2),oo(3,4)) 1 1 ATRealSet_true.
AlgEquiv Pass %union(oc(1,2),co(2,3)) oo(1,3) 1 1 ATRealSet_true.
AlgEquiv Pass %union(oc(1,2),co(2,3)) cc(1,3) 0 0 ATRealSet_false.
AlgEquiv Pass {-1,1} %union({-1,1}) 1 1 ATRealSet_true.
AlgEquiv Pass {1,3} cc(1,3) 0 0 ATRealSet_false.
AlgEquiv Pass %intersection(oc(-1,1),co(1,2)) %union({1}) 1 1 ATRealSet_true.
AlgEquiv Pass oo(-inf,1) oo(-inf,1) 1 1 ATRealSet_true.
AlgEquiv Pass oo(-1,inf) oo(0,inf) 0 0 ATRealSet_false.
AlgEquiv Pass %union(oc(-inf,0),oo(-1,4)) oo(-inf,4) 1 1 ATRealSet_true.
AlgEquiv Pass %union(oo(-inf,1),oo(-1,inf)) oo(-inf,inf) 1 1 ATRealSet_true.
AlgEquiv Pass all oo(-inf,inf) 1 1 ATRealSet_true.
Complex numbers
AlgEquiv Pass a=b/%i %i*a=b 1 1 ATEquation_num_i
AlgEquiv Pass b/%i=a %i*a=b 1 1 ATEquation_num_i
AlgEquiv Pass b=a/%i %i*a=b 0 0 ATEquation_lhs_notrhs_op
AlgEquiv Pass a*(2+%i)=b a=b/(2+%i) 1 1 ATEquation_ratio
AlgEquiv Pass a*(2+%i)=b a=b*(2-%i)/5 1 1 ATEquation_num_i
AlgEquiv Pass a*(2+%i)=b a=b*(2-%i)/4 0 0 ATEquation_default
Absolute value in equations
AlgEquiv Pass abs(x)=abs(y) x=y 0 0 ATEquation_default
AlgEquiv Pass abs(x)=abs(y) x=y or x=-y 1 1
AlgEquiv Pass abs(x)=abs(y) (x-y)*(x+y)=0 1 1
Functions
AlgEquiv Expected failure f(x):=1/0 f(x):=x^2 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
TEST_FAILED
AlgEquiv Pass 1 f(x):=x^2 0 0
Your answer should be a function, defined using the operator :=, but is not.
ATAlgEquiv_SA_not_function.
AlgEquiv Pass f(x)=x^2 f(x):=x^2 0 0
Your answer should be a function, defined using the operator :=, but is not.
ATAlgEquiv_SA_not_function.
AlgEquiv Pass f(x):=x^2 f(x,y):=x^2+y^2 0 0 ATFunction_length_args. ATFunction_false.
AlgEquiv Pass f(x):=x^2 f(x)=x^2 0 0
Your answer should be an equation, but is not.
ATAlgEquiv_SA_not_equation.
AlgEquiv Pass f(x):=x^2 f(x):=x^2 1 1 ATFunction_true.
AlgEquiv Pass f(x):=x^2 f(x):=sin(x) 0 0 ATFunction_false.
AlgEquiv Pass g(x):=x^2 f(x):=x^2 0 0 ATFunction_wrongname. ATFunction_true.
AlgEquiv Pass f(y):=y^2 f(x):=x^2 1 1 ATFunction_arguments_different. ATFunction_true.
AlgEquiv Pass f(a,b):=a^2+b^2 f(x,y):=x^2+y^2 1 1 ATFunction_arguments_different. ATFunction_true.
Inequalities
AlgEquiv Pass 1 x>1 0 0
Your answer should be an inequality, but is not.
ATAlgEquiv_SA_not_inequality.
AlgEquiv Pass x=1 x>1 and x<5 0 0
You have entered an equation, but an equation is not expected here. You may have typed something like "y=2*x+1" when you only needed to type "2*x+1".
ATAlgEquiv_TA_not_equation.
AlgEquiv Pass x<1 x>1 0 0
Your inequality appears to be backwards.
ATInequality_backwards.
AlgEquiv Pass 1<x x>1 1 1
AlgEquiv Pass a<b b>a 1 1
AlgEquiv Pass 2<2*x x>1 1 1
AlgEquiv Pass -2>-2*x x>1 1 1
AlgEquiv Pass x>1 x<=1 0 0
Your inequality should not be strict! Your inequality appears to be backwards.
ATInequality_strict. ATInequality_backwards.
AlgEquiv Pass x>=2 x<2 0 0
Your inequality should be strict, but is not! Your inequality appears to be backwards.
ATInequality_nonstrict. ATInequality_backwards.
AlgEquiv Pass x>=1 x>2 0 0
Your inequality should be strict, but is not!
ATInequality_nonstrict.
AlgEquiv Pass x>1 x>1 1 1
AlgEquiv Pass x>=1 x>=1 1 1
AlgEquiv Pass x>2 x>1 0 0
AlgEquiv Pass 1<x x>1 1 1
AlgEquiv Pass 2*x>=x^2 x^2<=2*x 1 1
AlgEquiv Pass 2*x>=x^2 x^2<=2*x 1 1
AlgEquiv Pass 3*x^2<9*a x^2-3*a<0 1 1
AlgEquiv Pass x^2>4 x>2 or x<-2 1 1 ATLogic_True.
AlgEquiv Pass 1<x or x<-3 x<-3 or 1<x 1 1 ATLogic_True.
AlgEquiv Pass 1<x or x<-3 x<-1 or 3<x 0 0
AlgEquiv Pass x>1 and x<5 x>1 and x<5 1 1 ATLogic_True.
AlgEquiv Pass x>1 and x<5 5>x and 1<x 1 1 ATLogic_True.
AlgEquiv Pass not (x<=2 and -2<=x) x>2 or -2>x 1 1 ATLogic_True.
AlgEquiv Pass x>2 or -2>x not (x<=2 and -2<=x) 1 1 ATLogic_True.
AlgEquiv Pass x>=1 or 1<=x x>=1 1 1
AlgEquiv Pass x>=1 and x<=1 x=1 1 1 ATInequality_solver.
AlgEquiv Pass (x>4 and x<5) or (x<-4 and x>-5) or (x+5>0 and x<-4) (x>-5 and x<-4) or (x>4 and x<5) 1 1 ATLogic_True.
AlgEquiv Pass (x>4 and x<5) or (x<-4 and x>-5) or (x+5>0 and x<-4) (x>-5 and x<-4) or (x>8 and x<5) 0 0
AlgEquiv Pass (x < 0 nounor x >= 1) nounand x <= 3 x < 0 or (x >= 1 and x <= 3) 1 1 ATLogic_True.
AlgEquiv Pass (x < 0 nounor x >= 1) nounand x <= 3 x < 0 or x >= 1 and x <= 3 1 1 ATLogic_True.
AlgEquiv Pass (x < 0 nounor x >= 1) nounand x <= 3 x < 0 or (x >= 1 and x <= 3) 1 1 ATLogic_True.
AlgEquiv Pass (x < 0 nounor x >= 1) nounand x <= 3 (x < 0 or x >= 1) and x <= 3 1 1 ATLogic_True.
AlgEquiv Pass (x < 0 nounor x >= 1) nounand x <= 3 x < 0 or (x >= 1 and x <= 3) 1 1 ATLogic_True.
AlgEquiv Pass natural_domain(1/x^2) natural_domain(1/x) 1 1 ATRealSet_true.
AlgEquiv Pass x^4>=0 x^2>=0 1 1
AlgEquiv Pass x^4>=16 x^2>=4 1 1
AlgEquiv Pass 2*x^2+x>=6 x<=-2 or x>=3/2 1 1 ATLogic_True.
AlgEquiv Pass {2,-2} x>2 nounor -2>x 0 0
Your answer should be an equation, inequality or a logical combination of many of these, but is not.
ATAlgEquiv_SA_not_logic.
AlgEquiv Pass x^2<4 x<2 nounand x>-2 1 1 ATLogic_Solver_True.
AlgEquiv Pass x^2<6 x<2 nounand x>-2 0 0
AlgEquiv Pass x>1 nounand x<-1 false 1 1 ATLogic_Solver_True.
AlgEquiv Pass x>1 nounand x<3 true 0 0
AlgEquiv Pass x>1 nounor x<3 true 1 1 ATLogic_Solver_True.
AlgEquiv Pass x>1 nounor x<3 all 1 1 ATLogic_Solver_True.
AlgEquiv Pass abs(x)<1 abs(x)<1 1 1
AlgEquiv Pass abs(x)<1 abs(x)<2 0 0
AlgEquiv Pass abs(x)<1 abs(x)>1 0 0
Your inequality appears to be backwards.
ATInequality_backwards.
AlgEquiv Pass abs(x)<2 -2<x and x<2 0 -3
AlgEquiv Pass -2<x and x<2 abs(x)<2 0 -3
AlgEquiv Pass abs(x)<2 -1<x and x<1 0 0
AlgEquiv Pass x^2<=9 abs(x)<3 0 0
AlgEquiv Pass x^2<=9 abs(x)<=3 0 -3
AlgEquiv Pass x^6<1 abs(x)<1 0 -3
AlgEquiv Pass abs(x)>1 x<-1 or x>1 0 -3
Surds
AlgEquiv Pass sqrt(12) 2*sqrt(3) 1 1
AlgEquiv Pass sqrt(11+6*sqrt(2)) 3+sqrt(2) 1 1
AlgEquiv Pass (19601-13860*sqrt(2))^(7/4) (5*sqrt(2)-7)^7 1 1
AlgEquiv Pass (19601-13861*sqrt(2))^(7/4) (5*sqrt(2)-7)^7 0 0
AlgEquiv Pass (19601-13861*sqrt(2))^(7/4) (5*sqrt(2)-7)^7 0 0
AlgEquiv Pass sqrt(2*log(26)+4-2*log(2)) sqrt(2*log(13)+4) 1 1
AlgEquiv Pass sqrt(2)*sqrt(3)+2*(sqrt(2/3))*x-(2/3)*(sqrt(2/3))*x^2+(4/9)*(sqrt(2/3))*x^3 4*sqrt(6)*x^3/27-(2*sqrt(6)*x^2)/9+(2*sqrt(6)*x)/3+sqrt(6) 1 1
Factorials and binomials
AlgEquiv Pass (n+1)*n! (n+1)! 1 1
AlgEquiv Pass n/n! 1/(n-1)! 1 1
AlgEquiv Pass n/n! 1/(n+1)! 0 0
AlgEquiv Pass n!/(k!*(n-k)!) binomial(n,k) 1 1
AlgEquiv Pass binomial(n,k)+binomial(n,k+1) binomial(n+1,k+1) 0 -3
AlgEquiv Pass binomial(n,k)+binomial(n,k+1) binomial(n+1,k) 0 0
AlgEquiv Pass binomial(n,k) binomial(n,n-k) 1 1
AlgEquiv Pass 175!*56!/(55!*176!) 17556/55176 1 1
Unevaluated derviatives
AlgEquiv Pass 3*s*diff(q(s),s) 3*s*diff(q(s),s) 1 1
Sums and products
AlgEquiv Pass sum(k^n,n,0,3) sum(k^n,n,0,3) 1 1
AlgEquiv Pass 1+k+k^2+k^3 sum(k^n,n,0,3) 1 1
AlgEquiv Pass 1+k+k^2 sum(k^n,n,0,3) 0 0
AlgEquiv Pass product(cos(k*x),k,1,3) product(cos(k*x),k,1,3) 1 1
AlgEquiv Pass cos(x)*cos(2*x)*cos(3*x) product(cos(k*x),k,1,3) 1 1
AlgEquiv Pass cos(x)*cos(2*x) product(cos(k*x),k,1,3) 0 0
Scientific units are ignored
AlgEquiv Pass 9.81*m/s^2 stackunits(9.81,m/s^2) 1 1
AlgEquiv Pass 6*stackunits(1,m) stackunits(6,m) 1 1
AlgEquiv Pass stackunits(2,m)^2 stackunits(4,m^2) 1 1
AlgEquiv Pass stackunits(2,s)^2 stackunits(4,m^2) 0 0
Maxima does not simplify -inf (I agree!)
AlgEquiv Pass -inf minf 0 0
These currently fail
AlgEquiv Pass 2/%i*ln(sqrt((1+z)/2)+%i*sqrt((1-z)/2)) -%i*ln(z+%i*sqrt(1-z^2)) 0 -3
AlgEquiv Pass abs(x^2-4)/(abs(x-2)*abs(x+2)) 1 0 -3
AlgEquiv Pass abs(x^2-4) abs(x-2)*abs(x+2) 0 -3
AlgEquiv Pass (-1)^n*cos(x)^n (-cos(x))^n 0 -3
AlgEquiv Pass (sqrt(108)+10)^(1/3)-(sqrt(108)-10)^(1/3) 2 0 -3
AlgEquiv Pass (sqrt(2+sqrt(2))+sqrt(2-sqrt(2)))/(2*sqrt(2)) sqrt(sqrt(2)+2)/2 0 -3
AlgEquiv Pass sqrt(2*x*sqrt(x^2+1)+2*x^2+1)-sqrt(x^2+1)-x 0 0 -3
AlgEquiv Pass (77+20*sqrt(13))^(1/6)-(77-20*sqrt(13))^(1/6) 1 0 -3
AlgEquiv Pass (930249+416020*sqrt(5))^(1/30)-(930249-416020*sqrt(5))^(1/30) 1 0 -3
AlgEquiv Pass cos(2*%pi/17) (-1+sqrt(17)+sqrt(34-2*sqrt(17)))/16+(2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17))))/16 0 -3
AlgEquiv Pass (41-sqrt(511))/2 (sqrt((4*(cos((1/2*(acos((61/1040*sqrt(130)))-atan(11/3)))))^(2))+21)-(2*cos((1/2*(acos((61/1040*sqrt(130)))-atan(11 / 3))))))^(2) 0 -3
AlgEquiv Pass a*(1+sqrt(2))=b a=b*(sqrt(2)-1)/3 0 -3 ATEquation_default
This is only equivalent for x>=0...
AlgEquiv Pass atan(1/2) %pi/2-atan(2) 0 -3
This is true for all x...
AlgEquiv Pass asinh(x) ln(x+sqrt(x^2+1)) 0 -3
Logical expressions
AlgEquiv Pass true and false false 1 1 ATLogic_True.
AlgEquiv Pass true or false false 0 0
AlgEquiv Pass A and B B and A 1 1 ATLogic_True.
AlgEquiv Pass A and B C and A 0 0
AlgEquiv Pass A and B=C C=B and A 1 1 ATLogic_True.
AlgEquiv Pass A and (B and C) A and B and C 1 1 ATLogic_True.
AlgEquiv Pass A and (B or C) A and (B or C) 1 1 ATLogic_True.
AlgEquiv Pass (A and B) or (A and C) A and (B or C) 1 1 ATLogic_True.
AlgEquiv Pass -(b#pm#sqrt(b^2-4*a*c)) -b#pm#sqrt(b^2-4*a*c) 1 1 ATLogic_True.
AlgEquiv Pass x=-b#pm#c^2 x=c^2-b or x=-c^2-b 1 1 ATLogic_True.
AlgEquiv Pass x#pm#a = y#pm#b x#pm#a = y#pm#b 1 1 ATEquation_sides
AlgEquiv Pass x#pm#a = y#pm#b x#pm#a = y#pm#c 0 0 ATEquation_lhs_notrhs
AlgEquiv Pass not(A) and not(B) not(A or B) 1 1 ATLogic_True.
AlgEquiv Pass not(A) and not(B) not(A and B) 0 0
AlgEquiv Pass not(A) or B boolean_form(A implies B) 1 1
AlgEquiv Pass not(A) or B A implies B 1 1 ATLogic_True.
AlgEquiv Pass not(A) and B A implies B 0 0
AlgEquiv Pass (not A and B) or (not B and A) A xor B 1 1 ATLogic_True.
AlgEquiv Pass (A and B) or (not A and not B) A xnor B 1 1 ATLogic_True.
AlgEquiv Pass {not(A) or B,A and B} {A implies B,A and B} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{{\rm not}\left( A \right)\,{\mbox{ or }}\, B \right \}\]
ATSet_wrongentries.
AlgEquiv Pass {A implies B,A and B} {not(A) and B,A and B} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{A\,{\mbox{ implies }}\, B \right \}\]
ATSet_wrongentries.
Differential equations
AlgEquiv Pass diff(x^2,x) 2*x 1 1
AlgEquiv Pass diff(x^2,x) 'diff(x^2,x) 1 1
AlgEquiv Pass noundiff(x^2,x) 2*x 1 1
AlgEquiv Pass diff(y,x) 0 1 1
AlgEquiv Pass noundiff(y,x) 0 1 1
AlgEquiv Pass diff(y(x),x) 0 0 0
Basic support for strings
AlgEquiv Pass "Hello" "Hello" 1 1 ATAlgEquiv_String
AlgEquiv Pass "hello" "Hello" 0 0 ATAlgEquiv_String
AlgEquiv Pass W "Hello" 0 0
Your answer should be a string, but is not.
ATAlgEquiv_SA_not_string.
AlgEquiv Pass "Hello" x^2 0 0
Your answer should be an expression, not an equation, inequality, list, set or matrix.
ATAlgEquiv_SA_not_expression.
AlgEquivNouns Expected failure 1/0 1 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATAlgEquivNouns_STACKERROR_SAns.
AlgEquivNouns Expected failure 1 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATAlgEquivNouns_STACKERROR_TAns.
AlgEquivNouns Expected failure (x-1)^2 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question.
ATAlgEquivNounsTEST_FAILED-Empty SA.
AlgEquivNouns Expected failure x^2 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty teacher answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty teacher answer, probably a CAS validation problem when authoring the question.
ATAlgEquivNounsTEST_FAILED-Empty TA.
AlgEquivNouns Expected failure x-1)^2 (x-1)^2 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question.
ATAlgEquivNounsTEST_FAILED-Empty SA.
AlgEquivNouns Pass diff(x^2,x) 2*x 1 1
AlgEquivNouns Pass diff(x^2,x) 'diff(x^2,x) 0 0
AlgEquivNouns Pass diff(x^2,x) 'diff(x^2,x) 0 0
AlgEquivNouns Pass 'diff(y,x) noundiff(y,x) 1 1
AlgEquivNouns Pass diff(y,x) 0 1 1
AlgEquivNouns Pass 'diff(y,x) 0 0 0
AlgEquivNouns Pass noundiff(y,x) 0 0 0
AlgEquivNouns Pass diff(y(x),x) 0 0 0
Differential equations
AlgEquivNouns Pass noundiff(H,x,2) = -R/T noundiff(H,x,2) + R/T = 0 1 1 ATEquation_ratio
AlgEquivNouns Pass 'diff(H,x,2) = -R/T noundiff(H,x,2) + R/T = 0 1 1 ATEquation_ratio
AlgEquivNouns Pass y(t)=int(s^2,s,0,t) y(t)=t^3/3 1 1 ATEquation_sides
AlgEquivNouns Pass y(t)='int(s^2,s,0,t) y(t)=t^3/3 0 0 ATEquation_lhs_notrhs
AlgEquivNouns Pass y(t)='int(s^2,s,0,t) y(t)=nounint(s^2,s,0,t) 1 1 ATEquation_sides
Logic nouns are still evaluated
AlgEquivNouns Pass true nounand false false 1 1 ATLogic_True.
SubstEquiv Expected failure 1/0 x^2-2*x+1 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSubstEquiv_STACKERROR_SAns.
SubstEquiv Expected failure x^2 x^2-2*x+1 [1/0] TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSubstEquiv_STACKERROR_Opt.
SubstEquiv Expected failure x^2 x^2-2*x+1 x 0 -1
The option to this answer test must be a list. This is an error. Please contact your teacher.
ATSubstEquiv_Opt_List.
SubstEquiv Pass x^2+1 x^2+1 1 1
SubstEquiv Pass x^2+1 x^3+1 0 0
SubstEquiv Pass x^2+1 x^3+1 0 0
SubstEquiv Pass X^2+1 x^2+1 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ X=x \right] \]
ATSubstEquiv_Subst [X = x].
SubstEquiv Pass x^2+y a^2+b 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ x=a , y=b \right] \]
ATSubstEquiv_Subst [x = a,y = b].
SubstEquiv Pass x^2+y/z a^2+c/b 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ x=a , y=c , z=b \right] \]
ATSubstEquiv_Subst [x = a,y = c,z = b].
SubstEquiv Pass y=x^2 a^2=b 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ x=a , y=b \right] \]
ATSubstEquiv_Subst [x = a,y = b].
SubstEquiv Pass {x=1,y=2} {x=2,y=1} 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ x=y , y=x \right] \]
ATSubstEquiv_Subst [x = y,y = x].
Where a variable is also a function name.
SubstEquiv Pass cos(a*x)/(x*(ln(x))) cos(a*y)/(y*(ln(y))) 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ a=a , x=y \right] \]
ATSubstEquiv_Subst [a = a,x = y].
SubstEquiv Pass cos(a*x)/(x*(ln(x))) cos(x*a)/(a*(ln(a))) 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ a=x , x=a \right] \]
ATSubstEquiv_Subst [a = x,x = a].
SubstEquiv Pass cos(a*x)/(x*(ln(x))) cos(a*x)/(x(ln(x))) 0 0
SubstEquiv Pass cos(a*x)/(x*(ln(x))) cos(a*y)/(y(ln(y))) 0 0
SubstEquiv Pass x+1>y y+1>x 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ x=y , y=x \right] \]
ATSubstEquiv_Subst [x = y,y = x].
SubstEquiv Pass x+1>y x<y+1 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ x=y , y=x \right] \]
ATSubstEquiv_Subst [x = y,y = x].
Matrices
SubstEquiv Pass matrix([1,A^2+A+1],[2,0]) matrix([1,x^2+x+1],[2,0]) 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=x \right] \]
ATSubstEquiv_Subst [A = x].
SubstEquiv Pass matrix([B,A^2+A+1],[2,C]) matrix([y,x^2+x+1],[2,z]) 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=x , B=y , C=z \right] \]
ATSubstEquiv_Subst [A = x,B = y,C = z].
SubstEquiv Pass matrix([B,A^2+A+1],[2,C]) matrix([y,x^2+x+1],[2,x]) 0 0
The entries underlined in red below are those that are incorrect. \[ \left[\begin{array}{cc} {\color{red}{\underline{B}}} & {\color{red}{\underline{A^2+A+1}}} \\ 2 & {\color{red}{\underline{C}}} \end{array}\right]\]
ATMatrix_wrongentries.
Lists
SubstEquiv Pass [x^2+1,x^2] [A^2+1,A^2] 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ x=A \right] \]
ATSubstEquiv_Subst [x = A].
SubstEquiv Pass [x^2-1,x^2] [A^2+1,A^2] 0 0
The entries underlined in red below are those that are incorrect. \[\left[ {\color{red}{\underline{x^2-1}}} , {\color{red}{\underline{x ^2}}} \right] \]
(ATList_wrongentries 1, 2).
SubstEquiv Pass [A,B,C] [B,C,A] 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=B , B=C , C=A \right] \]
ATSubstEquiv_Subst [A = B,B = C,C = A].
SubstEquiv Pass [A,B,C] [B,B,A] 0 0
The entries underlined in red below are those that are incorrect. \[\left[ {\color{red}{\underline{A}}} , B , {\color{red}{\underline{C }}} \right] \]
(ATList_wrongentries 1, 3).
SubstEquiv Pass [1,[A,B],C] [1,[a,b],C] 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=a , B=b , C=C \right] \]
ATSubstEquiv_Subst [A = a,B = b,C = C].
Sets
SubstEquiv Pass {x^2+1,x^2} {A^2+1,A^2} 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ x=A \right] \]
ATSubstEquiv_Subst [x = A].
SubstEquiv Pass {x^2-1,x^2} {A^2+1,A^2} 0 0
The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. \[\left \{x^2-1 , x^2 \right \}\]
ATSet_wrongentries.
SubstEquiv Pass {A+1,B^2,C} {B,C+1,A^2} 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=C , B=A , C=B \right] \]
ATSubstEquiv_Subst [A = C,B = A,C = B].
SubstEquiv Pass {1,{A,B},C} {1,{a,b},C} 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=a , B=b , C=C \right] \]
ATSubstEquiv_Subst [A = a,B = b,C = C].
SubstEquiv Pass A*cos(t)+B*sin(t) P*cos(t)+Q*sin(t) 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=P , B=Q , t=t \right] \]
ATSubstEquiv_Subst [A = P,B = Q,t = t].
SubstEquiv Pass A*cos(t)+B*sin(t) P*cos(x)+Q*sin(x) 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=P , B=Q , t=x \right] \]
ATSubstEquiv_Subst [A = P,B = Q,t = x].
Fix some variables.
SubstEquiv Pass A*cos(t)+B*sin(t) P*cos(x)+Q*sin(x) [x] 0 0
SubstEquiv Pass A*cos(t)+B*sin(t) P*cos(x)+Q*sin(x) [t] 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=P , B=Q , t=x \right] \]
ATSubstEquiv_Subst [A = P,B = Q,t = x].
SubstEquiv Pass A*cos(t)*e^x+B*sin(t)*e^x+C*sin(2*x)+D*cos(2*x) P*cos(t)*e^x+Q*sin(t)*e^x+R*sin(2*x)+S*cos(2*x) [x,t] 1 1
Your answer would be correct if you used the following substitution of variables. \[\left[ A=P , B=Q , C=R , D=S \right] \]
ATSubstEquiv_Subst [A = P,B = Q,C = R,D = S].
EqualComAss Expected failure 1/0 0 0 -1 ATEqualComAss_STACKERROR_SAns.
EqualComAss Expected failure 0 1/0 0 -1 ATEqualComAss_STACKERROR_TAns.
Numbers
EqualComAss Pass 2/4 1/2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 3^2 8 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass 3^2 9 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass cos(0) 1 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 4^(1/2) 2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 1/3^(1/2) (1/3)^(1/2) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass sqrt(3)/3 (1/3)^(1/2) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass sqrt(3) 3^(1/2) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 2*sqrt(2) sqrt(8) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 2*2^(1/2) sqrt(8) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass sqrt(2)/4 1/sqrt(8) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 1/sqrt(2) 2^(1/2)/2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 4.0 4 0 0 ATEqualComAss (AlgEquiv-true).
Case sensitivity
EqualComAss Pass X x 0 0 ATEqualComAss (AlgEquiv-false)ATAlgEquiv_WrongCase.
EqualComAss Pass exdowncase(X) x 1 1
EqualComAss Pass exdowncase((X-1)^2) x^2-2*x+1 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass exdowncase(X^2-2*X+1) x^2-2*x+1 1 1
Powers
EqualComAss Pass a^2/b^3 a^2*b^(-3) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass lg(a^x,a) x 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass x^(2/4) x^(1/2) 0 0 ATEqualComAss (AlgEquiv-true).
Simple polynomials
EqualComAss Pass 1+2*x x*2+1 1 1
EqualComAss Pass 1+x 2*x+1 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass 1+x+x 2*x+1 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (x+y)+z z+x+y 1 1
EqualComAss Pass x*x x^2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (x+5)*x x*(5+x) 1 1
EqualComAss Pass x*(x+5) 5*x+x^2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (1-x)^2 (x-1)^2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (a-x)^6000 (x-a)^6000 0 0 ATEqualComAss (AlgEquiv-true).
Expressions with subscripts
EqualComAss Pass rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) 1 1
EqualComAss Pass rho*z*V/(4*pi*epsilon[1]*(R^2+z^2)^(3/2)) rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) 0 0 ATEqualComAss (AlgEquiv-false).
Unary minus
EqualComAss Pass -1+2 2-1 1 1
EqualComAss Pass -1*2+3*4 3*4-1*2 1 1
EqualComAss Pass (-1*2)+3*4 10 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass -1*2+3*4 3*4-1*2 1 1
EqualComAss Pass x*(-y) -x*y 1 1
EqualComAss Pass x*(-y) -(x*y) 1 1
EqualComAss Pass (-x)*(-x) x*x 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (-x)*(-x) x^2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass -1/4*%pi*i -(%i*%pi)/4 0 0 ATEqualComAss (AlgEquiv-true).
Rational expressions
EqualComAss Pass 1/2 3/6 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 1/(1+2*x) 1/(2*x+1) 1 1
EqualComAss Pass 2/(4+2*x) 1/(x+2) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (a*b)/c a*(b/c) 1 1
EqualComAss Pass ((x+1)/(x*(x-1)))*(x-1) ((x+1)*(x-1))/(x*(x-1)) 1 1
EqualComAss Pass (-x)/y -(x/y) 1 1
EqualComAss Pass x/(-y) -(x/y) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass -1/(1-x) 1/(x-1) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 1/2*1/x 1/(2*x) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (k+8)/(k^2+4*k-12) (k+8)/(k^2+4*k-12) 1 1
EqualComAss Pass (k+8)/(k^2+4*k-12) (k+8)/((k-2)*(k+6)) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (k+7)/(k^2+4*k-12) (k+8)/(k^2+4*k-12) 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass -(2*k+6)/(k^2+4*k-12) -(2*k+6)/(k^2+4*k-12) 1 1
EqualComAss Pass (a+b)/1 (b+a)/1 1 1
No simplicifcation here
EqualComAss Pass 1*x x 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 23+0*x 23 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass x+0 x 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass x^1 x 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (1/2)*(a+b) (a+b)/2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 1/3*logbase(27,6) logbase(27,6)/3 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 1/3*lg(27,6) lg(27,6)/3 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass lg(root(x, n)) lg(x, 10)/n 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass 1/3*i i/3 0 0 ATEqualComAss (AlgEquiv-true).
Complex numbers
EqualComAss Pass %i e^(i*pi/2) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (4*sqrt(3)*%i+4)^(1/5) rectform((4*sqrt(3)*%i+4)^(1/5)) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (4*sqrt(3)*%i+4)^(1/5) 8^(1/5)*(cos(%pi/15)+%i*sin(%pi/15)) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass (4*sqrt(3)*%i+4)^(1/5) polarform((4*sqrt(3)*%i+4)^(1/5)) 0 0 ATEqualComAss (AlgEquiv-true).
Equations
EqualComAss Pass y=x x=y 1 1
EqualComAss Pass x+1 y=2*x+1 0 0
Your answer should be an equation, but is not.
ATEqualComAss ATAlgEquiv_SA_not_equation.
EqualComAss Pass y=1+2*x y=2*x+1 1 1
EqualComAss Pass y=x+x+1 y=1+2*x 0 0 ATEqualComAss (AlgEquiv-true).
Logic
EqualComAss Pass A and B B and A 1 1
EqualComAss Pass A or B B or A 1 1
EqualComAss Pass A or B B and A 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass not(true) false 0 0 ATEqualComAss (AlgEquiv-true).
Sets
EqualComAss Pass {2*x+1,2} {2, 1+x*2} 1 1
EqualComAss Pass 2 {2} 0 0
Your answer should be a set, but is not. Note that the syntax to enter a set is to enclose the comma separated values with curly brackets.
ATEqualComAss ATAlgEquiv_SA_not_set.
EqualComAss Pass {2*x+1, 1+1} {2, 1+x*2} 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass {1,2} {1,{2}} 0 0 ATEqualComAss (AlgEquiv-false)ATSet_wrongentries.
EqualComAss Pass {4,3} {3,4} 1 1
EqualComAss Pass {4,4} {4} 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass {-1,1,-1} {-1,-1,1} 1 1
EqualComAss Pass {-1,1,-1} {-1,1} 0 0 ATEqualComAss (AlgEquiv-true).
Lists
EqualComAss Pass [2*x+1,2] [1+x*2,2] 1 1
EqualComAss Pass [x+x+1, 1+1] [1+x*2,2] 0 0 ATEqualComAss (AlgEquiv-true).
Matrices
EqualComAss Pass matrix([1,2],[2,3]) matrix([1,2],[2,3]) 1 1
EqualComAss Pass matrix([1,2],[2,3]) matrix([1,2,3],[2,3,3]) 0 0 ATEqualComAss (AlgEquiv-false)ATMatrix_wrongsz_columns.
EqualComAss Pass matrix([1,2],[2,3]) matrix([1,2],[2,5]) 0 0 ATEqualComAss (AlgEquiv-false)ATMatrix_wrongentries.
EqualComAss Pass matrix([1,2],[2,2+1]) matrix([1,2],[2,3]) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass matrix([x+x, 1],[1, 1]) matrix([2*x, 1],[1, 1]) 0 0 ATEqualComAss (AlgEquiv-true).
Sums and products
EqualComAss Pass sum(k^n,n,0,3) sum(k^n,n,0,3) 1 1
EqualComAss Pass 1+k+k^2+k^3 sum(k^n,n,0,3) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass product(cos(k*x),k,1,3) product(cos(k*x),k,1,3) 1 1
EqualComAss Pass cos(x)*cos(2*x)*cos(3*x) product(cos(k*x),k,1,3) 0 0 ATEqualComAss (AlgEquiv-true).
Inequalities are not commutative under this test
EqualComAss Pass -6/5 > x x < -6/5 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass x<1 and -3<x -3<x and x<1 1 1
EqualComAss Pass 1>x and -3<x -3<x and x<1 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass make_less_ineq(-6/5 > x) x < -6/5 1 1
EqualComAss Pass make_less_ineq(1>x and -3<x) -3<x and x<1 1 1
EqualComAss Pass make_less_ineq(6/3 > x) x < 2 0 0 ATEqualComAss (AlgEquiv-true).
Unary Equations
EqualComAss Pass 1 stackeq(1) 1 1
EqualComAss Pass stackeq(1) 1 1 1
EqualComAss Pass stackeq(1+1) 2 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass stackeq(1) 0 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass lowesttermsp(1/3) true 1 1
EqualComAss Pass lowesttermsp(2/6) true 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass lowesttermsp(x^2/x) true 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass lowesttermsp(-y/-x) true 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass lowesttermsp((x^2-1)/(x-1)) true 0 0 ATEqualComAss (AlgEquiv-false).
EqualComAss Pass lowesttermsp((x^2-1)/(x+2)) true 1 1
Bad things in denominators
EqualComAss Pass rationalized(1+sqrt(3)/3) true 1 1
EqualComAss Pass rationalized(1+1/sqrt(3)) [sqrt(3)] 1 1
EqualComAss Pass rationalized(1/sqrt(3)) [sqrt(3)] 1 1
EqualComAss Pass rationalized(1/sqrt(2)+i/sqrt(2)) [sqrt(2),sqrt(2)] 1 1
EqualComAss Pass rationalized(sqrt(2)/2+1/sqrt(3)) [sqrt(3)] 1 1
EqualComAss Pass rationalized(1/sqrt(2)+1/sqrt(3)) [sqrt(2),sqrt(3)] 1 1
EqualComAss Pass rationalized(1/(1+i)) [i] 1 1
EqualComAss Pass rationalized(1/(1+1/root(3,2))) [root(3,2)] 1 1
Differential Equations
EqualComAss Pass diff(y,x) 0 1 1
EqualComAss Pass diff(x^2,x) 2*x 1 1
EqualComAss Pass noundiff(x^2,x) 2*x 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass diff(y,x) 'diff(y,x) 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAss Pass noundiff(y,x) 'diff(y,x) 1 1
EqualComAss Pass 'diff(y(x),x) 'diff(y(x),x,1) 1 1
EqualComAss Pass noundiff(y(x),x)=-x/4 4*noundiff(y(x),x)+x=0 0 0 ATEqualComAss (AlgEquiv-true).
EqualComAssRules Expected failure 1/0 0 [] 0 -1 ATEqualComAssRules_STACKERROR_SAns.
EqualComAssRules Expected failure 0 1/0 [] 0 -1 ATEqualComAssRules_STACKERROR_TAns.
EqualComAssRules Expected failure 0+a a TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Missing option when executing the test.
STACKERROR_OPTION.
EqualComAssRules Expected failure 0+a a x 0 -1
The option to this answer test must be a non-empty list of supported rules. This is an error. Please contact your teacher.
ATEqualComAssRules_Opt_List.
EqualComAssRules Expected failure 0+a a [x] 0 -1
The option to this answer test must be a non-empty list of supported rules. This is an error. Please contact your teacher.
ATEqualComAssRules_Opt_Wrong.
EqualComAssRules Expected failure 0+a a [intMul,intFac] 0 -1
The option to this answer test contains incompatible rules. This is an error. Please contact your teacher.
ATEqualComAssRules_Opt_Incompatible.
Basic cases
EqualComAssRules Pass 1+1 3 [zeroAdd] 0 0 ATEqualComAssRules (AlgEquiv-false).
EqualComAssRules Pass 1+1 2 [zeroAdd] 0 0
EqualComAssRules Pass 1+1 2 [testdebug,zeroAdd] 0 0 ATEqualComAssRules: [1 noun+ 1,2].
EqualComAssRules Pass 0+a a [zeroAdd] 1 1
EqualComAssRules Pass a+0 a [zeroAdd] 1 1
EqualComAssRules Pass 1*a a [testdebug,zeroAdd] 0 0 ATEqualComAssRules: [1 noun* a,a].
EqualComAssRules Pass 1*a a [oneMul] 1 1
EqualComAssRules Pass 1*a a ID_TRANS 1 1
EqualComAssRules Pass a/1 a ID_TRANS 1 1
EqualComAssRules Pass 0*a 0 ID_TRANS 1 1
EqualComAssRules Pass 0-1*i -i ID_TRANS 1 1
EqualComAssRules Pass 0-i -i ID_TRANS 1 1
EqualComAssRules Pass 2+1*i 2+i ID_TRANS 1 1
EqualComAssRules Pass x^0+x^1/1+x^2/2+x^3/3!+x^4/4! 1+x+x^2/2+x^3/3!+x^4/4! ID_TRANS 1 1
EqualComAssRules Pass 0^(1-1) 0 ID_TRANS 0 0 ATEqualComAssRules_STACKERROR_SAns.
EqualComAssRules Pass 0*a 0 delete(zeroMul, ID_TRANS) 0 0
EqualComAssRules Pass -(-a) a [negNeg] 1 1
EqualComAssRules Pass -(-(-a)) -a [negNeg] 1 1
EqualComAssRules Pass -(-(-a)) a [testdebug,negNeg] 0 0 ATEqualComAssRules (AlgEquiv-false).
EqualComAssRules Pass 3/(-x) -3/x ID_TRANS 0 0
EqualComAssRules Pass 3/(-x) -3/x [testdebug,ID_TRANS] 0 0 ATEqualComAssRules: [3 noun* UNARY_RECIP UNARY_MINUS noun* x,UNARY_MINUS noun* 3 noun* UNARY_RECIP x].
EqualComAssRules Pass -x*(x+1) x*(-x-1) [negDist] 1 1
EqualComAssRules Pass -x*(x-1) x*(1-x) NEG_TRANS 1 1
EqualComAssRules Pass -x*(x-1) x*(1-x) NEG_TRANS 1 1
EqualComAssRules Pass -5*x*(3-x) 5*x*(x-3) NEG_TRANS 1 1
EqualComAssRules Pass -x*(x-1)*(x+1) x*(x-1)*(-x-1) NEG_TRANS 1 1
EqualComAssRules Pass -x*(x-1)*(x+1) x*(1-x)*(x+1) NEG_TRANS 1 1
EqualComAssRules Pass -x*(y-1)*(x-1) x*(1-x)*(y-1) NEG_TRANS 1 1
EqualComAssRules Pass -x*(y-1)*(x-1) x*(x-1)*(1-y) NEG_TRANS 1 1
EqualComAssRules Pass (x-y)*(y-x) -(x-y)*(x-y) NEG_TRANS 1 1
EqualComAssRules Pass (x-y)*(y-x) -(x-y)^2 [testdebug,NEG_TRANS] 0 0 ATEqualComAssRules: [UNARY_MINUS noun* (x noun+ UNARY_MINUS noun* y) noun* (x noun+ UNARY_MINUS noun* y),UNARY_MINUS noun* (x noun+ UNARY_MINUS noun* y) noun^ 2].
EqualComAssRules Pass -x*(x-1)*(x+1) x*(1-x)*(x+1) [testdebug,negDist,negNeg] 0 0 ATEqualComAssRules: [x noun* (UNARY_MINUS noun* 1 noun+ UNARY_MINUS noun* x) noun* (x noun+ UNARY_MINUS noun* 1),x noun* (1 noun+ UNARY_MINUS noun* x) noun* (1 noun+ x)].
EqualComAssRules Pass -x*(y-1)*(x-1) x*(x-1)*(1-y) [testdebug,negDist,negNeg] 0 0 ATEqualComAssRules: [x noun* (1 noun+ UNARY_MINUS noun* x) noun* (y noun+ UNARY_MINUS noun* 1),x noun* (1 noun+ UNARY_MINUS noun* y) noun* (x noun+ UNARY_MINUS noun* 1)].
EqualComAssRules Pass 3/(-x) -3/x [negDiv] 1 1
EqualComAssRules Pass 3/(-x) ev(-3,simp)/x [negDiv] 1 1
EqualComAssRules Pass (-a)/(-x) -(-a/x) [testdebug,ID_TRANS] 0 0 ATEqualComAssRules: [UNARY_MINUS noun* a noun* UNARY_RECIP UNARY_MINUS noun* x,UNARY_MINUS noun* UNARY_MINUS noun* a noun* UNARY_RECIP x].
EqualComAssRules Pass (-a)/(-x) -(-a/x) [negDiv] 1 1
EqualComAssRules Pass (-a)/(-x) a/x [testdebug,negDiv] 0 0 ATEqualComAssRules: [UNARY_MINUS noun* UNARY_MINUS noun* a noun* UNARY_RECIP x,a noun* UNARY_RECIP x].
EqualComAssRules Pass (-a)/(-x) a/x [negDiv,negNeg] 1 1
EqualComAssRules Pass 1/(-x) (-1)/x [negDiv] 1 1
EqualComAssRules Pass 1/(-x) ev(-1,simp)/x [negDiv] 1 1
EqualComAssRules Pass (2/-3)*(x-y) -(2/3)*(x-y) [negDiv] 1 1
EqualComAssRules Pass (2/-3)*(x-y) (2/3)*(y-x) [negDiv] 0 0
EqualComAssRules Pass (2/-3)*(x-y) (2/3)*(y-x) [negDiv,negOrd] 1 1
EqualComAssRules Pass -2/(1-x) 2/(x-1) [testdebug,negDiv] 0 0 ATEqualComAssRules: [UNARY_MINUS noun* 2 noun* UNARY_RECIP (1 noun+ UNARY_MINUS noun* x),2 noun* UNARY_RECIP (x noun+ UNARY_MINUS noun* 1)].
EqualComAssRules Pass 1/2*3/x 3/(2*x) [testdebug,ID_TRANS] 0 0 ATEqualComAssRules: [3 noun* (UNARY_RECIP 2) noun* UNARY_RECIP x,3 noun* UNARY_RECIP 2 noun* x].
EqualComAssRules Pass 1/2*3/x 3/(2*x) [ID_TRANS,recipMul] 1 1
EqualComAssRules Pass 5/2*3/x 15/(2*x) [testdebug,ID_TRANS,recipMul] 0 0 ATEqualComAssRules: [3 noun* 5 noun* UNARY_RECIP 2 noun* x,15 noun* UNARY_RECIP 2 noun* x].
EqualComAssRules Pass -(x-y) y-x [negOrd] 1 1
EqualComAssRules Pass 5/2*3/x 15/(2*x) [ID_TRANS,recipMul,intMul] 1 1
EqualComAssRules Pass (3+2)*x+x 5*x+x [ID_TRANS,intAdd] 1 1
EqualComAssRules Pass (3-5)*x+x -2*x+x [ID_TRANS,intAdd] 1 1
EqualComAssRules Pass 7*x*(-3*x) -21*x*x [ID_TRANS,intMul] 1 1
EqualComAssRules Pass (-7*x)*(-3*x) 21*x*x [testdebug,ID_TRANS,intMul] 0 0 ATEqualComAssRules: [UNARY_MINUS noun* UNARY_MINUS noun* 21 noun* x noun* x,21 noun* x noun* x].
EqualComAssRules Pass (-7*x)*(-3*x) 21*x*x [ID_TRANS,intMul,negNeg] 1 1
ev(a/b/c, simp)=a/(b*c)
EqualComAssRules Pass a/b/c a/(b*c) [testdebug,ID_TRANS] 0 0 ATEqualComAssRules: [a noun* (UNARY_RECIP b) noun* UNARY_RECIP c,a noun* UNARY_RECIP b noun* c].
EqualComAssRules Pass a/b/c a/(b*c) [ID_TRANS,recipMul] 1 1
EqualComAssRules Pass (a/b)/c a/(b*c) [ID_TRANS,recipMul] 1 1
ev(a/(b/c), simp)=(a*c)/b
EqualComAssRules Pass a/(b/c) (a*c)/b [testdebug,ID_TRANS] 0 0 ATEqualComAssRules: [a noun* UNARY_RECIP b noun* UNARY_RECIP c,a noun* c noun* UNARY_RECIP b].
EqualComAssRules Pass a/(b/c) (a*c)/b [testdebug,ID_TRANS,recipMul] 0 0 ATEqualComAssRules: [a noun* UNARY_RECIP b noun* UNARY_RECIP c,a noun* c noun* UNARY_RECIP b].
EqualComAssRules Pass a/(b/c) (a*c)/b [ID_TRANS,divDiv] 1 1
EqualComAssRules Pass A*a/(B*b/c) A*(a*c)/(B*b) [ID_TRANS,divDiv] 1 1
EqualComAssRules Pass A*a/(B*b/c)*1/d A*(a*c)/(B*b)*1/d [ID_TRANS,divDiv] 1 1
EqualComAssRules Pass D*A*a/(B*b/c)*1/d A*(a*c)/(B*b)*D/d [ID_TRANS,divDiv] 1 1
EqualComAssRules Pass A*a/(B*b/c)*1/d A*(a*c)/(B*b*d) [testdebug,ID_TRANS,divDiv] 0 0 ATEqualComAssRules: [A noun* a noun* c noun* (UNARY_RECIP B noun* b) noun* UNARY_RECIP d,A noun* a noun* c noun* UNARY_RECIP B noun* b noun* d].
EqualComAssRules Pass A*a/(B*b/c)*1/d A*(a*c)/(B*b*d) [ID_TRANS,divDiv,recipMul] 1 1
EqualComAssRules Pass A/(B/(C/D)) A*C/(B*D) [testdebug,ID_TRANS,divDiv] 0 0 ATEqualComAssRules: [A noun* C noun* (UNARY_RECIP B) noun* UNARY_RECIP D,A noun* C noun* UNARY_RECIP B noun* D].
EqualComAssRules Pass A/(B/(C/D)) A*C/(B*D) [ID_TRANS,divDiv,recipMul] 1 1
EqualComAssRules Pass 18 2*3^2 [intFac] 1 1
CasEqual Expected failure 1/0 x^2-2*x+1 0 -1 ATCASEqual_STACKERROR_SAns.
CasEqual Expected failure x 1/0 0 -1 ATCASEqual_STACKERROR_TAns.
CasEqual Pass 0.5 1/2 x 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass x=1 1 0 0
You have entered an equation, but an equation is not expected here. You may have typed something like "y=2*x+1" when you only needed to type "2*x+1".
ATCASEqual ATAlgEquiv_TA_not_equation.
Case sensitivity
CasEqual Pass a A 0 0 ATCASEqual_false.
CasEqual Pass exdowncase(X^2-2*X+1) x^2-2*x+1 1 1 ATCASEqual_true.
Numbers
CasEqual Pass 4^(-1/2) 1/2 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass ev(4^(-1/2),simp) ev(1/2,simp) 1 1 ATCASEqual_true.
CasEqual Pass 2^2 4 0 0 ATCASEqual (AlgEquiv-true).
Powers
CasEqual Pass a^2/b^3 a^2*b^(-3) 0 0 ATCASEqual (AlgEquiv-true).
Expressions with subscripts
CasEqual Pass rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) 1 1 ATCASEqual_true.
CasEqual Pass rho*z*V/(4*pi*epsilon[1]*(R^2+z^2)^(3/2)) rho*z*V/(4*pi*epsilon[0]*(R^2+z^2)^(3/2)) 0 0 ATCASEqual_false.
Mix of floats and rational numbers
CasEqual Pass 0.5 1/2 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass x^(1/2) sqrt(x) 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass ev(x^(1/2),simp) ev(sqrt(x),simp) 1 1 ATCASEqual_true.
CasEqual Pass abs(x) sqrt(x^2) 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass ev(abs(x),simp) ev(sqrt(x^2),simp) 1 1 ATCASEqual_true.
CasEqual Pass x-1 (x^2-1)/(x+1) 0 0 ATCASEqual (AlgEquiv-true).
Polynomials and rational function
CasEqual Pass x+x 2*x 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass ev(x+x,simp) ev(2*x,simp) 1 1 ATCASEqual_true.
CasEqual Pass x+x^2 x^2+x 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass ev(x+x^2,simp) ev(x^2+x,simp) 1 1 ATCASEqual_true.
CasEqual Pass (x-1)^2 x^2-2*x+1 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass (x-1)^(-2) 1/(x^2-2*x+1) 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass 1/n-1/(n+1) 1/(n*(n+1)) 0 0 ATCASEqual (AlgEquiv-true).
Trig functions
CasEqual Pass cos(x) cos(-x) 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass ev(cos(x),simp) ev(cos(-x),simp) 1 1 ATCASEqual_true.
CasEqual Pass cos(x)^2+sin(x)^2 1 0 0 ATCASEqual (AlgEquiv-true).
CasEqual Pass 2*cos(x)^2-1 cos(2*x) 0 0 ATCASEqual (AlgEquiv-true).
Predicate function wrapper
CasEqual Pass imag_numberp(2*%i) true 1 1 ATCASEqual_true.
CasEqual Pass imag_numberp(%e^(%i*%pi/2)) true 1 1 ATCASEqual_true.
CasEqual Pass imag_numberp(2) false 1 1 ATCASEqual_true.
CasEqual Pass imag_numberp(%e^(%pi/2)) false 1 1 ATCASEqual_true.
CasEqual Pass complex_exponentialp(3*%e^(%i*%pi/6)) true 1 1 ATCASEqual_true.
CasEqual Pass complex_exponentialp(3) true 1 1 ATCASEqual_true.
CasEqual Pass complex_exponentialp(%e^(%i*%pi/6)) true 1 1 ATCASEqual_true.
CasEqual Pass complex_exponentialp(%e^%i) true 1 1 ATCASEqual_true.
CasEqual Pass complex_exponentialp(%e^(%pi/6)) true 1 1 ATCASEqual_true.
CasEqual Pass complex_exponentialp(3+%i) false 1 1 ATCASEqual_true.
CasEqual Pass complex_exponentialp(%e^(%i)/4) true 1 1 ATCASEqual_true.
CasEqual Pass complex_exponentialp(3*exp(%i*%pi/6)) true 1 1 ATCASEqual_true.
CasEqual Pass integerp(-1) true 0 0 ATCASEqual_false.
CasEqual Pass integerp(ev(-1,simp)) true 1 1 ATCASEqual_true.
SameType Expected failure 1/0 1 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSameType_STACKERROR_SAns.
SameType Expected failure 1 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSameType_STACKERROR_TAns.
Numbers
SameType Pass 4^(-1/2) 1/2 1 1
Lists
SameType Pass x [1,2,3] 0 0
SameType Pass [1,2] [1,2,3] 1 1
SameType Pass [1,x>2] [1,2<x] 1 1
SameType Pass [1,x,3] [1,2<x,4] 0 0
Sets
SameType Pass x {1,2,3} 0 0
SameType Pass {1,2} {1,2,3} 1 1
Matrices
SameType Pass matrix([1,2],[2,3]) matrix([1,2],[2,3]) 1 1
SameType Pass [[1,2],[2,3]] matrix([1,2],[2,3]) 0 0
SameType Pass matrix([1,2],[2,3]) matrix([1,2,3],[2,3,3]) 1 1
SameType Pass matrix([x>4,{1,x^2}],[[1,2],[1,3]]) matrix([4-x<0,{x^2, 1}],[[1,2],[1,3]]) 1 1
SameType Pass matrix([x>4,[1,x^2]],[[1,2],[1,3]]) matrix([4-x<0,{x^2, 1}],[[1,2],[1,4]]) 0 0
Equations
SameType Pass 1 x=1 0 0
SameType Pass x=1 x=1 1 1
Inequalities
SameType Pass 1 x>1 0 0
SameType Pass x>2 x>1 1 1
SameType Pass x>1 x>=1 1 1
SameType Pass x>1 and x<3 x>=1 1 1
SameType Pass {x>1,x<3} x>=1 0 0
SameType Pass sqrt(2)*sqrt(3)+2*(sqrt(2/3))*x-(2/3)*(sqrt(2/3))*x^2+(4/9)*(sqrt(2/3))*x^3 4*sqrt(6)*x^3/27-(2*sqrt(6)*x^2)/9+(2*sqrt(6)*x)/3+sqrt(6) 1 1
Basic tests
SysEquiv Expected failure 1/0 [(x-1)*(x+1)=0] TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSysEquiv_STACKERROR_SAns.
SysEquiv Expected failure [(x-1)*(x+1)=0] 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSysEquiv_STACKERROR_TAns.
SysEquiv Pass 1 [(x-1)*(x+1)=0] 0 0
Your answer should be a list, but it is not!
ATSysEquiv_SA_not_list.
SysEquiv Pass [(x-1)*(x+1)=0] 1 0 0
The teacher's answer is not a list. Please contact your teacher.
ATSysEquiv_SB_not_list.
SysEquiv Pass [1] [90=v*t,90=(v+5)*(t-1/4)] 0 0
Your answer should be a list of equations, but it is not!
ATSysEquiv_SA_not_eq_list.
SysEquiv Pass [(x-1)*(x+1)=0] [1] 0 0
The teacher's answer is not a list of equations, but should be.
ATSysEquiv_SB_not_eq_list.
SysEquiv Pass [x^2] [(x-1)*(x+1)=0] 0 0
Your answer should be a list of equations, but it is not!
ATSysEquiv_SA_not_eq_list.
SysEquiv Pass [90=v*t^t,90=(v+5)*(t-1/4)] [90=v*t,90=(v+5)*(t-1/4)] 0 0
One or more of your equations is not a polynomial!
ATSysEquiv_SA_not_poly_eq_list.
SysEquiv Pass [90=v*t,90=(v+5)*(t-1/4)] [90=v*t^t,90=(v+5)*(t-1/4)] 0 0
The Teacher's answer should be a list of polynomial equations, but is not. Please contact your teacher.
ATSysEquiv_SB_not_poly_eq_list.
Tests of equivalence
SysEquiv Pass [x^2=1] [(x-1)*(x+1)=0] 1 1
SysEquiv Pass [x^2+y^2=4,y=x] [y=x,y^2=2] 1 1
SysEquiv Pass [x^2+y^2=2,y=x] [y=x,y^2=2] 0 0
The entries underlined in red below are those that are incorrect. \[\left[ {\color{red}{\underline{y^2+x^2=2}}} , y=x \right] \]
ATSysEquiv_SA_system_overdetermined.
SysEquiv Pass [x=1] [(x-1)*(x+1)=0,(x-1)*(x-3)=0] 1 1 ATSysEquiv_SA_Completely_solved.
SysEquiv Pass [3*a+b-c=2, a-b+2*c=5,b+c=5] [a=1,b=2,c=3] 1 1
SysEquiv Pass [a=1,b=2,c=3] [3*a+b-c=2, a-b+2*c=5,b+c=5] 1 1 ATSysEquiv_SA_Completely_solved.
SysEquiv Pass [x^2=1] [(x-1)*(x+1)*(x-2)=0] 0 0
The entries underlined in red below are those that are incorrect. \[\left[ {\color{red}{\underline{x^2=1}}} \right] \]
ATSysEquiv_SA_system_overdetermined.
SysEquiv Pass [x=1,y=-1] [(x-1)*(y+1)=0] 0 0 ATSysEquiv_SA_Not_completely_solved.
SysEquiv Pass [x=1] [(x-1)*(x+1)=0] 0 0 ATSysEquiv_SA_Not_completely_solved.
SysEquiv Pass [x=1] [(x-1)*(x+1)*y=0] 0 0 ATSysEquiv_SA_Not_completely_solved.
SysEquiv Pass [90=v*t,90=(v+5)*(t-1/4)] [90=v*t,90=(v+5)*(t-1/4)] 1 1
SysEquiv Pass [90=v*t,90=(v+5)*(t*x-1/4)] [90=v*t,90=(v+5)*(t-1/4)] 0 0
Your answer includes too many variables!
ATSysEquiv_SA_extra_variables.
SysEquiv Pass [90=v*t,90=(v+5)*(t-1/4)] [90=v*t,90=(v+5)*(t*x-1/4)] 0 0
Your answer is missing one or more variables!
ATSysEquiv_SA_missing_variables.
SysEquiv Pass [90=v*t] [90=v*t,90=(v+5)*(t-1/4)] 0 0
The equations in your system appear to be correct, but you need others besides.
ATSysEquiv_SA_system_underdetermined.
SysEquiv Pass [90=v*t,90=(v+5)*(t-1/4),90=(v+6)*(t-1/5)] [90=v*t,90=(v+5)*(t-1/4)] 0 0
The entries underlined in red below are those that are incorrect. \[\left[ 90=t\cdot v , 90=\left(t-\frac{1}{4}\right)\cdot \left(v+5 \right) , {\color{red}{\underline{90=\left(t-\frac{1}{5}\right) \cdot \left(v+6\right)}}} \right] \]
ATSysEquiv_SA_system_overdetermined.
SysEquiv Pass [90=v*t,90=(v+5)*(t-1/4),90=(v+6)*(t-1/5),90=(v+7)*(t-1/4),90=(v+8)*(t-1/3)] [90=v*t,90=(v+5)*(t-1/4)] 0 0
The entries underlined in red below are those that are incorrect. \[\left[ 90=t\cdot v , 90=\left(t-\frac{1}{4}\right)\cdot \left(v+5 \right) , {\color{red}{\underline{90=\left(t-\frac{1}{5}\right) \cdot \left(v+6\right)}}} , {\color{red}{\underline{90=\left(t- \frac{1}{4}\right)\cdot \left(v+7\right)}}} , {\color{red} {\underline{90=\left(t-\frac{1}{3}\right)\cdot \left(v+8\right)}}} \right] \]
ATSysEquiv_SA_system_overdetermined.
Wrong variables
SysEquiv Pass [b^2=a,a=9] [x^2=y,y=9] 0 0
Your answer uses the wrong variables!
ATSysEquiv_SA_wrong_variables.
SysEquiv Pass [x^2=4] [x^2=4,y=9] 0 0
Your answer is missing one or more variables!
ATSysEquiv_SA_missing_variables.
SysEquiv Pass [d=90,d=v*t,d=(v+5)*(t-1/4)] [90=v*t,90=(v+5)*(t-1/4)] 0 0
Your answer includes too many variables!
ATSysEquiv_SA_extra_variables.
SysEquiv Pass stack_eval_assignments([d=90,d=v*t,d=(v+5)*(t-1/4)]) [90=v*t,90=(v+5)*(t-1/4)] 1 1
Sets Expected failure {1/0} {0} TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSets_STACKERROR_SAns.
Sets Expected failure {0} {1/0} TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSets_STACKERROR_TAns.
Sets Pass x {1,2,3} 0 0
Your answer should be a set, but is not. Note that the syntax to enter a set is to enclose the comma separated values with curly brackets.
ATSets_SA_not_set.
Sets Pass {1,2} x 0 0
The "Sets" answer test expects its second argument to be a set. This is an error. Please contact your teacher.
ATSets_SB_not_set.
Sets Pass {1,2} {1,2,3} 0 0
The following are missing from your set. \[\left \{3 \right \}\]
ATSets_missingentries.
Sets Pass {1,2,4} {1,2} 0 0
These entries should not be elements of your set. \[\left \{4 \right \}\]
ATSets_wrongentries.
Sets Pass {1,2,2+2} {1,2} 0 0
These entries should not be elements of your set. \[\left \{4 \right \}\]
ATSets_wrongentries.
Sets Pass {5,1,2,4} {1,2,3} 0 0
These entries should not be elements of your set. \[\left \{4 , 5 \right \}\] The following are missing from your set. \[\left \{3 \right \}\]
ATSets_wrongentries. ATSets_missingentries.
Sets Pass {2/4, 1/3} {1/2, 1/3} 1 1
Duplicate entries
Sets Pass {1,2,1} {1,2} 1 1
Your set appears to contain duplicate entries!
ATSets_duplicates.
Sets Pass {1,2,1+1} {1,2} 1 1
Your set appears to contain duplicate entries!
ATSets_duplicates.
Sets Pass {1,2,1+1} {1,2,3} 0 0
Your set appears to contain duplicate entries! The following are missing from your set. \[\left \{3 \right \}\]
ATSets_duplicates. ATSets_missingentries.
Sets Pass {(x-a)^6000} {(a-x)^6000} 0 0
These entries should not be elements of your set. \[\left \{{\left(x-a\right)}^{6000} \right \}\] The following are missing from your set. \[\left \{{\left(a-x\right)}^{6000} \right \}\]
ATSets_wrongentries. ATSets_missingentries.
Expanded Expected failure 1/0 0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATExpanded_STACKERROR_SAns.
Expanded Pass x>2 x^2-2*x+1 0 0
Your answer should be an expression, not an equation, inequality, list, set or matrix.
ATExpanded_SA_not_expression.
Expanded Pass x^2-1 0 1 1 ATExpanded_TRUE.
Expanded Pass 2*(x-1) 0 0 0 ATExpanded_FALSE.
Expanded Pass (x-1)*(x+1) 0 0 0 ATExpanded_FALSE.
Expanded Pass (x-a)*(x-b) 0 0 0 ATExpanded_FALSE.
Expanded Pass x^2-(a+b)*x+a*b 0 0 0 ATExpanded_FALSE.
Expanded Pass x^2-a*x-b*x+a*b 0 1 1 ATExpanded_TRUE.
Expanded Pass cos(2*x) 0 1 1 ATExpanded_TRUE.
Expanded Pass p+1 0 1 1 ATExpanded_TRUE.
Expanded Pass (p+1)*(p-1) 0 0 0 ATExpanded_FALSE.
Expanded Pass 3+2*sqrt(3) 0 1 1 ATExpanded_TRUE.
Expanded Pass 3+sqrt(12) 0 1 1 ATExpanded_TRUE.
Expanded Pass (1+sqrt(5))*(1-sqrt(3)) 0 0 0 ATExpanded_FALSE.
This fails, but you are never going to ask students to do this anyway...
Expanded Expected failure (maths) (a-x)^6000 0 1 -2 ATExpanded_TRUE.
FacForm Expected failure 1/0 0 x 0 -1 ATFacForm_STACKERROR_SAns.
FacForm Expected failure 0 1/0 x 0 -1 ATFacForm_STACKERROR_TAns.
FacForm Expected failure 0 0 1/0 0 -1 ATFacForm_STACKERROR_Opt.
Trivial cases
FacForm Pass 2 2 x 1 1 ATFacForm_int_true.
FacForm Pass 6 6 x 1 1 ATFacForm_int_true.
FacForm Pass 1/3 1/3 x 1 1 ATFacForm_true.
FacForm Pass 3*x^2 3*x^2 x 1 1 ATFacForm_true.
FacForm Pass 4*x^2 4*x^2 x 1 1 ATFacForm_true.
Linear integer factors
FacForm Pass 2*(x-1) 2*x-2 x 1 1 ATFacForm_true.
FacForm Pass 2*x-2 2*x-2 x 0 0
Your answer is not factored. You need to take out a common factor.
ATFacForm_notfactored.
FacForm Pass 2*(x+1) 2*x-2 x 0 0
Your answer is factored, well done. Note that your answer is not algebraically equivalent to the correct answer. You must have done something wrong.
ATFacForm_isfactored. ATFacForm_default_true. ATFacForm_notalgequiv.
FacForm Pass 2*x+2 2*x-2 x 0 0
Your answer is not factored. You need to take out a common factor. Note that your answer is not algebraically equivalent to the correct answer. You must have done something wrong.
ATFacForm_notfactored. ATFacForm_notalgequiv.
FacForm Pass 2*(x+0.5) 2*x+1 x 1 1 ATFacForm_default_true.
Linear factors
FacForm Pass t*(2*x+1) t*(2*x+1) x 1 1 ATFacForm_true.
FacForm Pass t*x+t t*(x+1) x 0 0
Your answer is not factored. This term is expected to be a polynomial, but is not.
ATFacForm_notfactored.
Quadratic, with no const
FacForm Pass 2*x*(x-3) 2*x^2-6*x x 1 1 ATFacForm_true.
FacForm Pass 2*(x^2-3*x) 2*x*(x-3) x 0 0
Your answer is not factored. You could still do some more work on the term \(x^2-3\cdot x\).
ATFacForm_notfactored.
FacForm Pass x*(2*x-6) 2*x*(x-3) x 0 0
Your answer is not factored. You could still do some more work on the term \(2\cdot x-6\). You need to take out a common factor.
ATFacForm_notfactored.
Quadratic
FacForm Pass (x+2)*(x+3) (x+2)*(x+3) x 1 1 ATFacForm_true.
FacForm Pass (x+2)*(2*x+6) 2*(x+2)*(x+3) x 0 0
Your answer is not factored. You could still do some more work on the term \(2\cdot x+6\). You need to take out a common factor.
ATFacForm_notfactored.
FacForm Pass (z*x+z)*(2*x+6) 2*z*(x+1)*(x+3) x 0 0
Your answer is not factored. You could still do some more work on the term \(z\cdot x+z\). This term is expected to be a polynomial, but is not. You could still do some more work on the term \(2\cdot x+6\). You need to take out a common factor.
ATFacForm_notfactored.
FacForm Pass (x+t)*(x-t) x^2-t^2 x 1 1 ATFacForm_true.
FacForm Pass t^2-1 (t-1)*(t+1) t 0 0
Your answer is not factored.
ATFacForm_notfactored.
FacForm Pass t^2+1 t^2+1 t 1 1 ATFacForm_true.
FacForm Pass v^2+1 v^2+1 v 1 1 ATFacForm_true.
FacForm Pass v^2-1 v^2-1 v 0 0
Your answer is not factored.
ATFacForm_notfactored.
FacForm Pass -(3*w-4*v+9*u)*(3*w+4*v-u) -(3*w-4*v+9*u)*(3*w+4*v-u) v 1 1 ATFacForm_true.
FacForm Pass -(6*k*(4*b-k-1)) 6*k*(1+k-4*b) k 1 1 ATFacForm_true.
FacForm Pass -(6*a*(4*b-a-1)) 6*a*(1+a-4*b) a 1 1 ATFacForm_true.
FacForm Pass -(6*a*(4*b-a-1)) 6*a*(-(4*b)+a+1) a 1 1 ATFacForm_true.
FacForm Pass x*(x-4+4/x) x^2-4*x+4 x 0 0
Your answer is not factored. You could still do some more work on the term \(x-4+\frac{4}{x}\). This term is expected to be a polynomial, but is not.
ATFacForm_notfactored.
These are delicate cases!
FacForm Pass (2-x)*(3-x) (x-2)*(x-3) x 1 1 ATFacForm_true.
FacForm Pass (1-x)^2 (x-1)^2 x 1 1 ATFacForm_true.
FacForm Pass (1-x)*(1-x) (x-1)^2 x 1 1 ATFacForm_true.
FacForm Pass -(1-x)^2 -(x-1)^2 x 1 1 ATFacForm_true.
FacForm Pass (1-x)^2 (x-1)^2 x 1 1 ATFacForm_true.
FacForm Pass 4*(1-x/2)^2 (x-2)^2 x 1 1 ATFacForm_default_true.
FacForm Pass -3*(x-4)*(x+1) -3*x^2+9*x+12 x 1 1 ATFacForm_true.
FacForm Pass 3*(-x+4)*(x+1) -3*x^2+9*x+12 x 1 1 ATFacForm_true.
FacForm Pass 3*(4-x)*(x+1) -3*x^2+9*x+12 x 1 1 ATFacForm_true.
Cubics
FacForm Pass (x-1)*(x^2+x+1) x^3-1 x 1 1 ATFacForm_true.
FacForm Pass x^3-x+1 x^3-x+1 x 1 1 ATFacForm_true.
FacForm Pass 7*x^3-7*x+7 7*(x^3-x+1) x 0 0
Your answer is not factored. You need to take out a common factor.
ATFacForm_notfactored.
FacForm Pass (1-x)*(2-x)*(3-x) -x^3+6*x^2-11*x+6 x 1 1 ATFacForm_true.
FacForm Pass (2-x)*(2-x)*(3-x) -x^3+7*x^2-16*x+12 x 1 1 ATFacForm_true.
FacForm Pass (2-x)^2*(3-x) -x^3+7*x^2-16*x+12 x 1 1 ATFacForm_true.
FacForm Pass (x^2-4*x+4)*(3-x) -x^3+7*x^2-16*x+12 x 0 0
Your answer is not factored. You could still do some more work on the term \(x^2-4\cdot x+4\).
ATFacForm_notfactored.
FacForm Pass (x^2-3*x+2)*(3-x) -x^3+6*x^2-11*x+6 x 0 0
Your answer is not factored. You could still do some more work on the term \(x^2-3\cdot x+2\).
ATFacForm_notfactored.
FacForm Pass 3*y^3-6*y^2-24*y 3*(y-4)*y*(y+2) y 0 0
Your answer is not factored. You need to take out a common factor.
ATFacForm_notfactored.
FacForm Pass 3*(y^3-2*y^2-8*y) 3*(y-4)*y*(y+2) y 0 0
Your answer is not factored. You could still do some more work on the term \(y^3-2\cdot y^2-8\cdot y\).
ATFacForm_notfactored.
FacForm Pass 3*y*(y^2-2*y-8) 3*(y-4)*y*(y+2) y 0 0
Your answer is not factored. You could still do some more work on the term \(y^2-2\cdot y-8\).
ATFacForm_notfactored.
FacForm Pass 3*(y^2-4*y)*(y+2) 3*(y-4)*y*(y+2) y 0 0
Your answer is not factored. You could still do some more work on the term \(y^2-4\cdot y\).
ATFacForm_notfactored.
FacForm Pass (y-4)*y*(3*y+6) 3*(y-4)*y*(y+2) y 0 0
Your answer is not factored. You could still do some more work on the term \(3\cdot y+6\). You need to take out a common factor.
ATFacForm_notfactored.
FacForm Pass (a-x)^6000 (a-x)^6000 x 1 1 ATFacForm_true.
FacForm Pass (x-a)^6000 (a-x)^6000 x 1 1 ATFacForm_true.
Needs flattening
FacForm Pass 2*a*(a*b-1) 2*a*(a*b-1) a 1 1 ATFacForm_true.
FacForm Pass (2*a)*(a*b-1) 2*a*(a*b-1) a 1 1 ATFacForm_true.
Not polynomials in a variable
FacForm Pass (sin(x)+1)*(sin(x)-1) sin(x)^2-1 sin(x) 1 1 ATFacForm_true.
FacForm Pass (cos(t)-sqrt(2))^2 cos(t)^2-2*sqrt(2)*cos(t)+2 cos(t) 1 1 ATFacForm_true.
FacForm Pass 7 7 x 1 1 ATFacForm_int_true.
Factors over other fields
FacForm Pass 24*(x-1/4) 24*x-6 x 1 1 ATFacForm_default_true.
FacForm Pass (x-sqrt(2))*(x+sqrt(2)) x^2-2 x 1 1 ATFacForm_true.
FacForm Pass x^2-2 x^2-2 x 1 1 ATFacForm_true.
FacForm Pass (%i*x-2*%i) %i*(x-2) x 0 0
Your answer is not factored.
ATFacForm_notfactored.
FacForm Pass %i*(x-2) (%i*x-2*%i) x 1 1 ATFacForm_true.
FacForm Pass (x-%i)*(x+%i) x^2+1 x 1 1 ATFacForm_true.
FacForm Pass (x-1)*(x+(1+sqrt(3)*%i)/2)*(x+(1-sqrt(3)*%i)/2) x^3-1 x 1 1 ATFacForm_default_true.
CompSquare Expected failure 1/0 0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Missing option when executing the test.
STACKERROR_OPTION.
CompSquare Expected failure 1/0 0 x TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATCompSquare_STACKERROR_SAns.
CompSquare Expected failure 0 1/0 x TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATCompSquare_STACKERROR_TAns.
CompSquare Expected failure 0 0 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATCompSquare_STACKERROR_Opt.
Category errors.
CompSquare Pass 1 (x-1)^2+1 x 0 0
Your answer should depend on the variable \(x\) but it does not!
ATCompSquare_SA_not_depend_var.
CompSquare Pass (t-1)^2+1 (x-1)^2+1 x 0 0
Your answer should depend on the variable \(x\) but it does not!
ATCompSquare_SA_not_depend_var.
CompSquare Pass (x-1)^2+1=0 (x-1)^2+1 x 0 0
Your answer should be an expression, not an equation, inequality, list, set or matrix.
ATCompSquare_STACKERROR_LIST.
CompSquare Pass sin(x-1)+a-1 (x-1)^2+1 x 0 0 ATCompSquare_false_not_AlgEquiv.
Trivial cases
CompSquare Pass 1 1 x 1 1 ATCompSquare_true_trivial.
CompSquare Pass x-a x-a x 1 1 ATCompSquare_true_trivial.
CompSquare Pass x^2 x^2 x 1 1 ATCompSquare_true.
CompSquare Pass x^2-1 (x-1)*(x+1) x 1 1 ATCompSquare_true.
CompSquare Pass (x-1)^2*k (x-1)^2*k x 1 1 ATCompSquare_true.
CompSquare Pass (x-1)^2/k (x-1)^2/k x 1 1 ATCompSquare_true.
Normal cases
CompSquare Pass (x-1)^2+1 (x-1)^2+1 x 1 1 ATCompSquare_true.
CompSquare Pass (1-x)^2+1 (x-1)^2+1 x 1 1 ATCompSquare_true.
CompSquare Pass (X-1)^2+1 (x-1)^2+1 x 0 0
Your answer should depend on the variable \(x\) but it does not!
ATCompSquare_SA_not_depend_var.
CompSquare Pass 9*(x-1)^2+1 (3*x-3)^2+1 x 1 1 ATCompSquare_true.
CompSquare Pass -(x-1)^2 -(x-1)^2 x 1 1 ATCompSquare_true.
CompSquare Pass -(1-x)^2 -(x-1)^2 x 1 1 ATCompSquare_true.
CompSquare Pass -(x-1)^2+3 -(x-1)^2+3 x 1 1 ATCompSquare_true.
CompSquare Pass -(1-x)^2+3 -(x-1)^2+3 x 1 1 ATCompSquare_true.
CompSquare Pass -4*(x-1)^2+3 -4*(x-1)^2+3 x 1 1 ATCompSquare_true.
CompSquare Pass -4*(x-1)^2+3 -(2*x-2)^2+3 x 1 1 ATCompSquare_true.
CompSquare Pass 3-4*(x-1)^2 -(2*x-2)^2+3 x 1 1 ATCompSquare_true.
CompSquare Pass (x-1)^2+1 (x+1)^2+1 x 0 0
Your answer appears to be in the correct form, but is not equivalent to the correct answer.
ATCompSquare_true_not_AlgEquiv.
CompSquare Pass (x-a^2)^2+1+b (x-a^2)^2+1+b x 1 1 ATCompSquare_true.
CompSquare Pass x^2-2*x+2 (x-1)^2+1 x 0 0
The completed square is of the form \( a(\cdots\cdots)^2 + b\) where \(a\) and \(b\) do not depend on your variable. More than one of your summands appears to depend on the variable in your answer.
ATCompSquare_false_no_summands.
CompSquare Pass x+1 (x-1)^2+1 x 0 0 ATCompSquare_false_not_AlgEquiv.
CompSquare Pass a*(x-1)^2+1 a*(x-1)^2+1 x 1 1 ATCompSquare_true.
CompSquare Pass -a*(x-1)^2+1 1-a*(x-1)^2 x 1 1 ATCompSquare_true.
Not simple variable
CompSquare Pass (sin(x)-1)^2+1 (sin(x)-1)^2+1 sin(x) 1 1 ATCompSquare_true.
CompSquare Pass (x^2-1)^2+1 (x^2-1)^2+1 x^2 1 1 ATCompSquare_true.
CompSquare Pass (y-1)^2+1 (y-1)^2+1 y 1 1 ATCompSquare_true.
CompSquare Pass (y+1)^2+1 (y-1)^2+1 y 0 0
Your answer appears to be in the correct form, but is not equivalent to the correct answer.
ATCompSquare_true_not_AlgEquiv.
CompSquare Pass (x-1)^2+1 (sin(x)-1)^2+1 sin(x) 0 0
Your answer should depend on the variable \({\it facdum}\) but it does not!
ATCompSquare_SA_not_depend_var.
PropLogic Expected failure 1/0 0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATPropLogic_STACKERROR_SAns.
PropLogic Expected failure 0 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATPropLogic_STACKERROR_TAns.
PropLogic Pass true true 1 1
PropLogic Pass true false 0 0
PropLogic Pass A implies B not(A) or B 1 1
PropLogic Pass (a and b and c) xor (a and b) xor (a and c) xor a xor true (a implies b) or c 1 1
Equiv Expected failure x [x^2=4,x=2 or x=-2] 0 -1
The first argument to the Equiv answer test should be a list, but the test failed. Please contact your teacher.
ATEquiv_SA_not_list.
Equiv Expected failure [x^2=4,x=2 or x=-2] x 0 -1
The second argument to the Equiv answer test should be a list, but the test failed. Please contact your teacher.
ATEquiv_SB_not_list.
Equiv Expected failure [1/0] [x^2=4,x=2 or x=-2] 0 -1 ATEquiv_STACKERROR_SAns.
Equiv Expected failure [x^2=4,x=2 or x=-2] [1/0] 0 -1 ATEquiv_STACKERROR_TAns.
Equiv Pass [x^2=4,x=2 or x=-2] [x^2=4,x=2 or x=-2] 1 1
\[\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [x^2=4,x=#pm#2,x=2 and x=-2] [x^2=4,x=2 or x=-2] 0 0
\[\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x= \pm 2& \cr \color{red}{\mbox{and/or confusion!}}&\left\{\begin{array}{l}x=2\cr x=-2\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR,ANDOR]
Equiv Pass [x^2=4,x=2] [x^2=4,x=2 or x=-2] 0 0
\[\begin{array}{lll} &x^2=4& \cr \color{red}{\Leftarrow}&x=2& \cr \end{array}\]
[EMPTYCHAR,IMPLIEDCHAR]
Equiv Pass [x^2=4,x=2] [x^2=4,x=2] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&x^2=4& \cr \color{green}{\Leftrightarrow}&x=2& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [x^2=4,x^2-4=0,(x-2)*(x+2)=0,x=2 or x=-2] [x^2=4,x=2 or x=-2] 1 1
\[\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x^2-4=0& \cr \color{green}{\Leftrightarrow}&\left(x-2\right)\cdot \left(x+2\right)=0& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2=4,x= #pm#2, x=2 or x=-2] [x^2=4,x=2 or x=-2] 1 1
\[\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x= \pm 2& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2-6*x+9=0,x=3] [x^2-6*x+9=0,x=3] 1 1
\[\begin{array}{lll} &x^2-6\cdot x+9=0& \cr \color{green}{\mbox{(Same roots)}}&x=3& \cr \end{array}\]
[EMPTYCHAR,SAMEROOTS]
EquivFirst Expected failure x [x^2=4,x=2 or x=-2] 0 -1
The first argument to the Equiv answer test should be a list, but the test failed. Please contact your teacher.
ATEquivFirst_SA_not_list.
EquivFirst Expected failure [x^2=4,x=2 or x=-2] x 0 -1
The second argument to the Equiv answer test should be a list, but the test failed. Please contact your teacher.
ATEquivFirst_SB_not_list.
EquivFirst Expected failure [1/0] [x^2=4,x=2 or x=-2] 0 -1 ATEquivFirst_STACKERROR_SAns.
EquivFirst Expected failure [x^2=4,x=2 or x=-2] [1/0] 0 -1 ATEquivFirst_STACKERROR_TAns.
EquivFirst Pass [x^2=4,x=2 or x=-2] [x^2=4,x=2 or x=-2] 1 1
\[\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
EquivFirst Pass [x^2=9,x=3 or x=-3] [x^2=4,x=2 or x=-2] 0 0
The first line in your argument must be "\(x^2=4\)".
ATEquivFirst_SA_wrong_start
EquivFirst Pass [x^2=4,x=2] [x^2=4,x=2 or x=-2] 0 0
\[\begin{array}{lll} &x^2=4& \cr \color{red}{\Leftarrow}&x=2& \cr \end{array}\]
[EMPTYCHAR,IMPLIEDCHAR]
EquivFirst Pass [x^2=4,x^2-4=0,(x-2)*(x+2)=0,x=2 or x=-2] [x^2=4,x=2 or x=-2] 1 1
\[\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x^2-4=0& \cr \color{green}{\Leftrightarrow}&\left(x-2\right)\cdot \left(x+2\right)=0& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
EquivFirst Pass [x^2=4,x= #pm#2, x=2 or x=-2] [x^2=4,x=2 or x=-2] 1 1
\[\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x= \pm 2& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
EquivFirst Pass [x^2-6*x+9=0,x=3] [x^2-6*x+9=0,x=3] 1 1
\[\begin{array}{lll} &x^2-6\cdot x+9=0& \cr \color{green}{\mbox{(Same roots)}}&x=3& \cr \end{array}\]
[EMPTYCHAR,SAMEROOTS]
SingleFrac Expected failure 1/0 1/n 0 -1 ATSingleFrac_STACKERROR_SAns.
SingleFrac Expected failure 0 1/0 0 -1 ATSingleFrac_STACKERROR_TAns.
SingleFrac Pass x=3 2 0 0
Your answer should be an expression, not an equation, inequality, list, set or matrix.
ATSingleFrac_SA_not_expression.
SingleFrac Pass 3 3 1 1
SingleFrac Pass 3 2 0 0
Your answer is not algebraically equivalent to the correct answer. You must have done something wrong.
ATSingleFrac_ret_exp.
SingleFrac Pass 1/m 1/n 0 0
Your answer is not algebraically equivalent to the correct answer. You must have done something wrong.
ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac Pass 1/n 1/n 1 1 ATSingleFrac_true.
SingleFrac Pass a+1/2 (2*a+1)/2 0 0
Your answer needs to be a single fraction of the form \( {a}\over{b} \).
ATSingleFrac_part.
SingleFrac Pass a+1/2 (2*a+1)/2 0 0
Your answer needs to be a single fraction of the form \( {a}\over{b} \).
ATSingleFrac_part.
SingleFrac Pass 4/(x^2+2*x-24)+2/(x^2+4*x-12) (6*x-16)/(x^3-28*x+48) 0 0
Your answer needs to be a single fraction of the form \( {a}\over{b} \).
ATSingleFrac_part.
2 subtly different answers for the same question
SingleFrac Pass 2*(1/n) 2/n 0 0
Your answer needs to be a single fraction of the form \( {a}\over{b} \).
ATSingleFrac_part.
SingleFrac Pass 2/n 2/n 1 1 ATSingleFrac_true.
Simple Mistakes
SingleFrac Pass 2/(n+1) 1/(n+1) 0 0
Your answer is not algebraically equivalent to the correct answer. You must have done something wrong.
ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac Pass (2*n+1)/(n+2) 1/n 0 0
Your answer is not algebraically equivalent to the correct answer. You must have done something wrong.
ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac Pass (2*n)/(n*(n+2)) (2*n)/(n*(n+3)) 0 0
Your answer is not algebraically equivalent to the correct answer. You must have done something wrong.
ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac Pass (x-1)/(x^2-1) 1/(x+1) 1 1 ATSingleFrac_true.
Fractions within fractions
SingleFrac Pass (1/2)/(3/4) 2/3 0 0
Your answer contains fractions within fractions. You need to clear these and write your answer as a single fraction.
ATSingleFrac_div.
SingleFrac Pass (x-2)/4/(2/x^2) (x-2)*x^2/8 0 0
Your answer contains fractions within fractions. You need to clear these and write your answer as a single fraction.
ATSingleFrac_div.
SingleFrac Pass 1/(1-1/x) x/(x-1) 0 0
Your answer contains fractions within fractions. You need to clear these and write your answer as a single fraction.
ATSingleFrac_div.
SingleFrac Pass (1+1/a)/a (1+a)/a^2 0 0
Your answer contains fractions within fractions. You need to clear these and write your answer as a single fraction.
ATSingleFrac_div.
SingleFrac Pass a/(1+1/a) a^2/(1+a) 0 0
Your answer contains fractions within fractions. You need to clear these and write your answer as a single fraction.
ATSingleFrac_div.
SingleFrac Pass (1+2*b/a)/c (a+2*b)/(a*c) 0 0
Your answer contains fractions within fractions. You need to clear these and write your answer as a single fraction.
ATSingleFrac_div.
SingleFrac Pass c/(1+2*b/a) a*c/(a+2*b) 0 0
Your answer contains fractions within fractions. You need to clear these and write your answer as a single fraction.
ATSingleFrac_div.
SingleFrac Pass a*c/(a+2*b) a*c/(a+2*b) 1 1 ATSingleFrac_true.
Negative cases
SingleFrac Pass -1/2 -1/2 1 1 ATSingleFrac_true.
SingleFrac Pass -1/2 -1/3 0 0
Your answer is not algebraically equivalent to the correct answer. You must have done something wrong.
ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac Pass -(1/2) -1/2 1 1 ATSingleFrac_true.
SingleFrac Pass -a/b -a/b 1 1 ATSingleFrac_true.
SingleFrac Pass (-a)/b -a/b 1 1 ATSingleFrac_true.
SingleFrac Pass a/(-b) -a/b 1 1 ATSingleFrac_true.
SingleFrac Pass -(a/b) -a/b 1 1 ATSingleFrac_true.
SingleFrac Pass -(1/(n-1)) 1/(1-n) 1 1 ATSingleFrac_true.
SingleFrac Pass a/(-1-1/a) -a^2/(1+a) 0 0
Your answer contains fractions within fractions. You need to clear these and write your answer as a single fraction.
ATSingleFrac_div.
Surds in answers
SingleFrac Pass ((sqrt(5))^3 +6)/15 ((sqrt(5))^3 +6)/15 1 1 ATSingleFrac_true.
SingleFrac Pass 1/(1-sqrt(2)) 1/(1-sqrt(2)) 1 1 ATSingleFrac_true.
SingleFrac Pass ((sqrt(5))^3+6)/15 ((sqrt(5))^3+6)/15 1 1 ATSingleFrac_true.
SingleFrac Pass (5^(3/2)+6)/15 ((sqrt(5))^3+6)/15 1 1 ATSingleFrac_true.
PartFrac Expected failure 1/0 3*x^2 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Missing option when executing the test.
STACKERROR_OPTION.
PartFrac Expected failure 1/0 3*x^2 x TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATPartFrac_STACKERROR_SAns.
PartFrac Expected failure 0 0 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATPartFrac_STACKERROR_Opt.
PartFrac Expected failure 0 1/0 x TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATPartFrac_STACKERROR_TAns.
PartFrac Pass 1/n=0 1/n n 0 0
Your answer should be an expression, not an equation, inequality, list, set or matrix.
ATPartFrac_SA_not_expression.
PartFrac Pass 1/n {1/n} n 0 0
The answer test failed. Please contact your systems administrator
ATPartFrac_TA_not_expression.
Basic tests
PartFrac Pass 1/m 1/n n 0 0
The variables in your answer are different to those of the question, please check them.
ATPartFrac_diff_variables.
PartFrac Pass 1/n 1/n n 1 1 ATPartFrac_true.
A simple cases, linear factors in denominator
PartFrac Pass 1/(n+1)-1/n 1/(n+1)-1/n n 1 1 ATPartFrac_true.
PartFrac Pass 1/(n+1)+1/(1-n) 1/(n+1)-1/(n-1) n 1 1 ATPartFrac_true.
PartFrac Pass 1/(2*(n-1))-1/(2*(n+1)) 1/((n-1)*(n+1)) n 1 1 ATPartFrac_true.
PartFrac Pass 1/(2*(n+1))-1/(2*(n-1)) 1/((n-1)*(n+1)) n 0 0
Your answer as a single fraction is \(-\frac{1}{\left(n-1\right)\cdot \left(n+1\right)}\)
ATPartFrac_ret_expression.
PartFrac Pass -9/(x-2) + -9/(x+1) -9/(x-2) + -9/(x+1) x 1 1 ATPartFrac_true.
Irreducible quadratic in denominator
PartFrac Pass 1/(x-1)-(x+1)/(x^2+1) 2/((x-1)*(x^2+1)) x 1 1 ATPartFrac_true.
PartFrac Pass 1/(2*x-2)-(x+1)/(2*(x^2+1)) 1/((x-1)*(x^2+1)) x 1 1 ATPartFrac_true.
PartFrac Pass 1/(2*(x-1))+x/(2*(x^2+1)) 1/((x-1)*(x^2+1)) x 0 0
Your answer as a single fraction is \(\frac{2\cdot x^2-x+1}{2\cdot \left(x-1\right)\cdot \left(x^2+1 \right)}\)
ATPartFrac_ret_expression.
2 answers to the same question
PartFrac Pass 3/(x+1) + 3/(x+2) 3*(2*x+3)/((x+1)*(x+2)) x 1 1 ATPartFrac_true.
PartFrac Pass 3*(1/(x+1) + 1/(x+2)) 3*(2*x+3)/((x+1)*(x+2)) x 1 1 ATPartFrac_true.
Algebraically equivalent, but numerators of same order than denominator, ie not in partial fraction form.
PartFrac Pass 3*x*(1/(x+1) + 2/(x+2)) -12/(x+2)-3/(x+1)+9 x 0 0 ATPartFrac_false_degree.
PartFrac Pass (3*x+3)*(1/(x+1) + 2/(x+2)) 9-6/(x+2) x 0 0 ATPartFrac_false_degree.
PartFrac Pass n/(2*n-1)-(n+1)/(2*n+1) 1/(4*n-2)-1/(4*n+2) n 0 0 ATPartFrac_false_degree.
Correct Answer, Numerator > Denominator
PartFrac Pass 10/(x+3) - 2/(x+2) + x -2 (x^3 + 3*x^2 + 4*x +2)/((x+2)*(x+3)) x 1 1 ATPartFrac_true.
PartFrac Pass 2*x+1/(x+1)+1/(x-1) 2*x^3/(x^2-1) x 1 1 ATPartFrac_true.
Simple mistakes
PartFrac Pass 1/(n*(n-1)) 1/(n*(n-1)) n 0 0 ATPartFrac_false_factor.
PartFrac Pass 1/(n-1)-1/n^2 1/((n+1)*n) n 0 0
If your answer is written as a single fraction then the denominator would be \(\left(n-1\right)\cdot n^2\). In fact, it should be \(n\cdot \left(n+1\right)\).
ATPartFrac_denom_ret.
PartFrac Pass 1/(n-1)-1/n 1/(n-1)+1/n n 0 0
Your answer as a single fraction is \(\frac{1}{\left(n-1\right)\cdot n}\)
ATPartFrac_ret_expression.
PartFrac Pass 1/(x+1)-1/x 1/(x-1)+1/x x 0 0
Your answer as a single fraction is \(-\frac{1}{x\cdot \left(x+1\right)}\)
ATPartFrac_ret_expression.
PartFrac Pass 1/(n*(n+1))+1/n 2/n-1/(n+1) n 0 0 ATPartFrac_false_factor.
Different Variables
PartFrac Pass 2/(x+1)-1/(x+2) s/((s+1)*(s+2)) s 0 0
The variables in your answer are different to those of the question, please check them.
ATPartFrac_diff_variables.
Too many parts in the partial fraction
PartFrac Pass s/((s+1)^2) + s/(s+2) - 1/(s+1) s/((s+1)*(s+2)) s 0 0
If your answer is written as a single fraction then the denominator would be \({\left(s+1\right)}^2\cdot \left(s+2\right)\). In fact, it should be \(\left(s+1\right)\cdot \left(s+2\right)\).
ATPartFrac_denom_ret.
Too few parts in the partial fraction
PartFrac Pass s/(s+2) - 1/(s+1) s/((s+1)*(s+2)*(s+3)) s 0 0
If your answer is written as a single fraction then the denominator would be \(\left(s+1\right)\cdot \left(s+2\right)\). In fact, it should be \(\left(s+1\right)\cdot \left(s+2\right)\cdot \left(s+3\right)\).
ATPartFrac_denom_ret.
Addition and Subtraction errors
PartFrac Pass 1/(x+1) + 1/(x+2) 2/(x+1) + 1/(x+2) x 0 0
Your answer as a single fraction is \(\frac{2\cdot x+3}{\left(x+1\right)\cdot \left(x+2\right)}\)
ATPartFrac_ret_expression.
PartFrac Pass 1/(x+1) + 1/(x+2) 1/(x+1) + 2/(x+2) x 0 0
Your answer as a single fraction is \(\frac{2\cdot x+3}{\left(x+1\right)\cdot \left(x+2\right)}\)
ATPartFrac_ret_expression.
Denominator Error
PartFrac Pass 1/(x+1) + 1/(x+2) 1/(x+3) + 1/(x+2) x 0 0
Your answer as a single fraction is \(\frac{2\cdot x+3}{\left(x+1\right)\cdot \left(x+2\right)}\)
ATPartFrac_ret_expression.
PartFrac Pass (2*x+1)/(x^2+1)-2/(x-1) (2*x+1)/(x^2+1)-2/(x-1) x 1 1 ATPartFrac_true.
PartFrac Pass (-5/(x+3))+(16/(x+3)^2)-(2/(x+2))+4 (-5/(x+3))+(16/(x+3)^2)-(2/(x+2))+4 x 1 1 ATPartFrac_true.
Cubic in the denominator
PartFrac Pass (3*x^2-5)/((x-4)^2*x) (3*x^2-5)/((x-4)^2*x) x 0 0 ATPartFrac_false_factor.
PartFrac Pass -4/(16*x)+53/(16*(x-4))+43/(4*(x-4)^2) (3*x^2-5)/((x-4)^2*x) x 0 0
Your answer as a single fraction is \(\frac{49\cdot x^2-8\cdot x-64}{16\cdot {\left(x-4\right)}^2\cdot x}\)
ATPartFrac_ret_expression.
PartFrac Pass -5/(16*x)+53/(16*(x-4))+43/(4*(x-4)^2) (3*x^2-5)/((x-4)^2*x) x 1 1 ATPartFrac_true.
PartFrac Pass (5*x+6)/((x+1)*(x+5)^2) (5*x+6)/((x+1)*(x+5)^2) x 0 0 ATPartFrac_false_factor.
PartFrac Pass -1/(16*(x+5))+19/(4*(x+5)^2)+1/(16*(x+1)) (5*x+6)/((x+1)*(x+5)^2) x 1 1 ATPartFrac_true.
PartFrac Pass 5/(x*(x+3)*(5*x-2)) 5/(x*(x+3)*(5*x-2)) x 0 0 ATPartFrac_false_factor.
PartFrac Pass 125/(34*(5*x-2))+5/(51*(x+3))-5/(6*x) 5/(x*(x+3)*(5*x-2)) x 1 1 ATPartFrac_true.
PartFrac Pass (3*x^2-5)/((4*x-4)^2*x) (3*x^2-5)/((4*x-4)^2*x) x 0 0 ATPartFrac_false_factor.
PartFrac Pass -4/(16*x)+1/(2*(x-1))-1/(8*(x-1)^2) (3*x^2-5)/((4*x-4)^2*x) x 0 0
Your answer as a single fraction is \(\frac{2\cdot x^2-x-2}{8\cdot {\left(x-1\right)}^2\cdot x}\)
ATPartFrac_ret_expression.
PartFrac Pass -5/(16*x)+1/(2*(x-1))-1/(8*(x-1)^2) (3*x^2-5)/((4*x-4)^2*x) x 1 1 ATPartFrac_true.
Diff Expected failure 1/0 3*x^2 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Missing option when executing the test.
STACKERROR_OPTION.
Diff Expected failure 0 1/0 (x TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Option field is invalid. You have a missing right bracket ) in the expression: (x.
STACKERROR_OPTION.
Diff Expected failure 1/0 3*x^2 x 0 -1 ATDiff_STACKERROR_SAns.
Diff Expected failure 0 1/0 x 0 -1 ATDiff_STACKERROR_TAns.
Diff Expected failure 0 0 1/0 0 -1 ATDiff_STACKERROR_Opt.
Basic tests
Diff Pass 3*x^2 3*x^2 x 1 1 ATDiff_true.
Diff Pass 3*X^2 3*x^2 x 0 0 ATDiff_var_SB_notSA.
Diff Pass x^4/4 3*x^2 x 0 0
It looks like you have integrated instead!
ATDiff_int.
Diff Pass x^4/4+1 3*x^2 x 0 0
It looks like you have integrated instead!
ATDiff_int.
Diff Pass x^4/4+c 3*x^2 x 0 0
It looks like you have integrated instead!
ATDiff_int.
Diff Pass y=x^4/4 x^4/4 x 0 0
Your answer should be an expression, not an equation, inequality, list, set or matrix.
ATDiff_SA_not_expression.
Diff Pass x^4/4 y=x^4/4 x 0 0
Diff Pass y=x^4/4 y=x^4/4 x 0 0
Your answer should be an expression, not an equation, inequality, list, set or matrix.
ATDiff_SA_not_expression.
Diff Pass 6000*(x-a)^5999 6000*(x-a)^5999 x 1 1 ATDiff_true.
Diff Pass 5999*(x-a)^5999 6000*(x-a)^5999 x 0 0
Variable mismatch tests
Diff Pass y^2-2*y+1 x^2-2*x+1 x 0 0 ATDiff_var_SB_notSA.
Diff Pass x^2-2*x+1 y^2-2*y+1 x 0 0 ATDiff_var_SA_notSB.
Diff Pass y^2+2*y+1 x^2-2*x+1 z 0 0 ATDiff_var_notSASB_SAnceSB.
Diff Pass x^4/4 3*x^2 y 0 0
Edge cases
Diff Pass e^x+c e^x x 0 0
It looks like you have integrated instead!
ATDiff_int.
Diff Pass e^x+2 e^x x 0 0
It looks like you have integrated instead!
ATDiff_int.
Diff Expected failure n*x^n n*x^(n-1) x TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. TIMEDOUT
ATDiff_STACKERROR_SAns.
Diff Pass n*x^n (assume(n>0), n*x^(n-1)) x 0 0
Int Expected failure 1/0 1 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Missing option when executing the test.
STACKERROR_OPTION.
Int Expected failure 1/0 1 x 0 -1 ATInt_STACKERROR_SAns.
Int Expected failure 1 1/0 x 0 -1 ATInt_STACKERROR_TAns.
Int Expected failure 0 0 1/0 0 -1 ATInt_STACKERROR_Opt.
Int Expected failure 0 0 [x,1/0] 0 -1 ATInt_STACKERROR_Opt.
Int Expected failure 0 0 [x,NOCONST,1/0] 0 -1 ATInt_STACKERROR_Opt.
Basic tests
Int Pass x^3/3 x^3/3 x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass x^3/3+1 x^3/3 x 0 0
You need to add a constant of integration. This should be an arbitrary constant, not a number.
ATInt_const_int.
Int Pass x^3/3+c x^3/3 x 1 1 ATInt_true.
Int Pass x^3/3+c+1 x^3/3 x 1 1 ATInt_true.
Int Pass x^3/3+3*c x^3/3 x 1 1 ATInt_true.
Int Pass x^3/3-c x^3/3 x 1 1 ATInt_true.
Int Pass x^3/3+c+k x^3/3 x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, you have a strange constant of integration. Please ask your teacher about this.
ATInt_weirdconst.
Int Pass x^3/3+c^2 x^3/3 x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, you have a strange constant of integration. Please ask your teacher about this.
ATInt_weirdconst.
Int Pass x^3/3*c x^3/3 x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[x^2\] In fact, the derivative of your answer, with respect to \(x\) is: \[c\cdot x^2\] so you must have done something wrong!
ATInt_generic.
Int Pass X^3/3+c x^3/3 x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[x^2\] In fact, the derivative of your answer, with respect to \(x\) is: \[0\] so you must have done something wrong!
ATInt_generic. ATInt_var_SB_notSA.
Int Pass sin(2*x) x^3/3 x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[x^2\] In fact, the derivative of your answer, with respect to \(x\) is: \[2\cdot \cos \left( 2\cdot x \right)\] so you must have done something wrong!
ATInt_generic.
Int Pass x^2/2-2*x+2+c (x-2)^2/2 x 1 1 ATInt_true.
Int Pass (t-1)^5/5+c (t-1)^5/5 t 1 1 ATInt_true.
Int Pass (v-1)^5/5+c (v-1)^5/5 v 1 1 ATInt_true.
Int Pass cos(2*x)/2+1+c cos(2*x)/2 x 1 1 ATInt_true.
Int Pass (x-a)^6001/6001+c (x-a)^6001/6001 x 1 1 ATInt_true.
Int Pass (x-a)^6001/6001 (x-a)^6001/6001 x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass 6000*(x-a)^5999 (x-a)^6001/6001 x 0 0
It looks like you have differentiated instead!
ATInt_diff.
Int Pass 4*%e^(4*x)/(%e^(4*x)+1) log(%e^(4*x)+1)+c x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[\frac{4\cdot e^{4\cdot x}}{e^{4\cdot x}+1}\] In fact, the derivative of your answer, with respect to \(x\) is: \[\frac{16\cdot e^{4\cdot x}}{e^{4\cdot x}+1}-\frac{16\cdot e^{8 \cdot x}}{{\left(e^{4\cdot x}+1\right)}^2}\] so you must have done something wrong!
ATInt_generic.
The teacher adds a constant
Int Pass x^3/3+c x^3/3+c x 1 1 ATInt_true.
Int Pass x^2/2-2*x+2+c (x-2)^2/2+k x 1 1 ATInt_true.
The teacher condones lack of constant, or numerical constant
Int Pass x^3/3 x^3/3 [x,NOCONST] 1 1 ATInt_const_condone.
Int Pass x^3/3+c x^3/3 [x,NOCONST] 1 1 ATInt_true.
Int Pass x^2/2-2*x+2 (x-2)^2/2+k [x,NOCONST] 1 1 ATInt_const_condone.
Int Pass x^3/3+1 x^3/3 [x,NOCONST] 1 1 ATInt_const_int_condone.
Int Pass x^3/3+c^2 x^3/3 [x,NOCONST] 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, you have a strange constant of integration. Please ask your teacher about this.
ATInt_weirdconst.
Int Pass n*x^n n*x^(n-1) x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[\left(n-1\right)\cdot n\cdot x^{n-2}\] In fact, the derivative of your answer, with respect to \(x\) is: \[n^2\cdot x^{n-1}\] so you must have done something wrong!
ATInt_generic.
Int Pass n*x^n (assume(n>0), n*x^(n-1)) x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[\left(n-1\right)\cdot n\cdot x^{n-2}\] In fact, the derivative of your answer, with respect to \(x\) is: \[n^2\cdot x^{n-1}\] so you must have done something wrong!
ATInt_generic.
Special case
Int Pass exp(x)+c exp(x) x 1 1 ATInt_true.
Int Pass exp(x) exp(x) x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass exp(x) exp(x) [x,NOCONST] 1 1 ATInt_const_condone.
Student differentiates by mistake
Int Pass 2*x x^3/3 x 0 0
It looks like you have differentiated instead!
ATInt_diff.
Int Pass 2*x+c x^3/3 x 0 0
It looks like you have differentiated instead!
ATInt_diff.
Sloppy logs (teacher ignores abs(x) )
Int Pass ln(x) ln(x) x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass ln(x) ln(x) [x,NOCONST] 1 1 ATInt_const_condone.
Int Pass ln(x)+c ln(x)+c x 1 1 ATInt_true_equiv.
Int Pass ln(k*x) ln(x)+c x 1 1 ATInt_true_equiv.
Fussy logs (teacher uses abs(x) )
Int Pass ln(x) ln(abs(x))+c x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass ln(x)+c ln(abs(x))+c x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass ln(x) ln(abs(x))+c [x, NOCONST] 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass ln(abs(x)) ln(abs(x))+c x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass ln(abs(x))+c ln(abs(x))+c x 1 1 ATInt_true_equiv.
Int Pass ln(k*x) ln(abs(x))+c x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass ln(k*abs(x)) ln(abs(x))+c x 1 1 ATInt_true_equiv.
Int Pass ln(abs(k*x)) ln(abs(x))+c x 1 1 ATInt_true_equiv.
Teacher uses ln(k*abs(x))
Int Pass ln(x) ln(k*abs(x)) x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass ln(x)+c ln(k*abs(x)) x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass ln(abs(x)) ln(k*abs(x)) x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass ln(abs(x))+c ln(k*abs(x)) x 1 1 ATInt_true_equiv.
Int Pass ln(k*x) ln(k*abs(x)) x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass ln(k*abs(x)) ln(k*abs(x)) x 1 1 ATInt_true_equiv.
Other logs
Int Pass ln(x)+ln(a) ln(k*abs(x+a)) x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_generic. ATInt_logabs.
Int Pass log(x)^2-2*log(c)*log(x)+k ln(c/x)^2 x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Please ask your teacher about this.
ATInt_EqFormalDiff.
Int Pass log(x)^2-2*log(c)*log(x)+k ln(abs(c/x))^2 x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[-\frac{2\cdot \ln \left( \frac{\left| c\right| }{\left| x\right| } \right)}{x}\] In fact, the derivative of your answer, with respect to \(x\) is: \[\frac{2\cdot \ln \left( x \right)}{x}-\frac{2\cdot \ln \left( c \right)}{x}\] so you must have done something wrong!
ATInt_generic.
Int Pass c-(log(2)-log(x))^2/2 -1/2*log(2/x)^2 x 1 1 ATInt_true_equiv.
Int Pass ln(abs(x+3))/2+c ln(abs(2*x+6))/2+c x 0 -3
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Please ask your teacher about this.
ATInt_EqFormalDiff.
Two logs
Int Pass log(abs(x-3))+log(abs(x+3)) log(abs(x-3))+log(abs(x+3)) x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass log(abs(x-3))+log(abs(x+3))+c log(abs(x-3))+log(abs(x+3)) x 1 1 ATInt_true_equiv.
Int Pass log(abs(x-3))+log(abs(x+3)) log(x-3)+log(x+3) x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass log(abs(x-3))+log(abs(x+3))+c log(x-3)+log(x+3) x 1 1 ATInt_true_equiv.
Int Pass log(x-3)+log(x+3) log(x-3)+log(x+3) x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass log(x-3)+log(x+3)+c log(x-3)+log(x+3) x 1 1 ATInt_true_equiv.
Int Pass log(x-3)+log(x+3) log(abs(x-3))+log(abs(x+3)) x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass log(x-3)+log(x+3)+c log(abs(x-3))+log(abs(x+3)) x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass log(abs((x-3)*(x+3)))+c log(abs(x-3))+log(abs(x+3)) x 1 1 ATInt_true_equiv.
Int Pass log(abs((x^2-9)))+c log(abs(x-3))+log(abs(x+3)) x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Please ask your teacher about this.
ATInt_EqFormalDiff.
Int Pass 2*log(abs(x-2))-log(abs(x+2))+(x^2+4*x)/2 -log(abs(x+2))+2*log(abs(x-2))+(x^2+4*x)/2+c x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass -log(abs(x+2))+2*log(abs(x-2))+(x^2+4*x)/2+c -log(abs(x+2))+2*log(abs(x-2))+(x^2+4*x)/2+c x 1 1 ATInt_true_equiv.
Int Pass -log(abs(x+2))+2*log(abs(x-2))+(x^2+4*x)/2+c -log((x+2))+2*log((x-2))+(x^2+4*x)/2 x 1 1 ATInt_true_equiv.
Inconsistent log(abs())
Int Pass log(abs(x-3))+log((x+3))+c log(x-3)+log(x+3) x 0 0
There appear to be strange inconsistencies between your use of \(\log(...)\) and \(\log(|...|)\). Please ask your teacher about this.
ATInt_true_equiv. ATInt_logabs_inconsistent.
Int Pass log((v-3))+log(abs(v+3))+c log(v-3)+log(v+3) v 0 0
There appear to be strange inconsistencies between your use of \(\log(...)\) and \(\log(|...|)\). Please ask your teacher about this.
ATInt_true_equiv. ATInt_logabs_inconsistent.
Int Pass log((x-3))+log(abs(x+3)) log(x-3)+log(x+3) x 0 0
There appear to be strange inconsistencies between your use of \(\log(...)\) and \(\log(|...|)\). Please ask your teacher about this.
ATInt_const. ATInt_logabs_inconsistent.
Int Pass 2*log((x-2))-log(abs(x+2))+(x^2+4*x)/2 -log(abs(x+2))+2*log(abs(x-2))+(x^2+4*x)/2 x 0 0
There appear to be strange inconsistencies between your use of \(\log(...)\) and \(\log(|...|)\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs. ATInt_logabs_inconsistent.
Significant integration constant differences
Int Pass 2*(sqrt(t)-5)-10*log((sqrt(t)-5))+c 2*(sqrt(t)-5)-10*log((sqrt(t)-5))+c t 1 1 ATInt_true_equiv.
Int Pass 2*(sqrt(t))-10*log((sqrt(t)-5))+c 2*(sqrt(t)-5)-10*log((sqrt(t)-5))+c t 1 1 ATInt_true_differentconst.
Int Pass 2*(sqrt(t)-5)-10*log((sqrt(t)-5))+c 2*(sqrt(t)-5)-10*log(abs(sqrt(t)-5))+c t 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass 2*(sqrt(t))-10*log(abs(sqrt(t)-5))+c 2*(sqrt(t)-5)-10*log(abs(sqrt(t)-5))+c t 1 1 ATInt_true_differentconst.
Trig
Int Pass 2*sin(x)*cos(x) sin(2*x)+c x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass 2*sin(x)*cos(x)+k sin(2*x)+c x 1 1 ATInt_true.
Int Pass -2*cos(3*x)/3-3*cos(2*x)/2 -2*cos(3*x)/3-3*cos(2*x)/2+c x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass -2*cos(3*x)/3-3*cos(2*x)/2+1 -2*cos(3*x)/3-3*cos(2*x)/2+c x 0 0
You need to add a constant of integration. This should be an arbitrary constant, not a number.
ATInt_const_int.
Int Pass -2*cos(3*x)/3-3*cos(2*x)/2+c -2*cos(3*x)/3-3*cos(2*x)/2+c x 1 1 ATInt_true.
Int Pass (tan(2*t)-2*t)/2 -(t*sin(4*t)^2-sin(4*t)+t*cos(4*t)^2+2*t*cos(4*t)+t)/(sin(4*t)^2+cos(4*t)^2+2*cos(4*t)+1) t 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass (tan(2*t)-2*t)/2+1 -(t*sin(4*t)^2-sin(4*t)+t*cos(4*t)^2+2*t*cos(4*t)+t)/(sin(4*t)^2+cos(4*t)^2+2*cos(4*t)+1) t 0 0
You need to add a constant of integration. This should be an arbitrary constant, not a number.
ATInt_const_int.
Int Pass (tan(2*t)-2*t)/2+c -(t*sin(4*t)^2-sin(4*t)+t*cos(4*t)^2+2*t*cos(4*t)+t)/(sin(4*t)^2+cos(4*t)^2+2*cos(4*t)+1) t 1 1 ATInt_true.
Int Pass tan(x)-x+c tan(x)-x x 1 1 ATInt_true.
Note the difference in feedback here, generated by the options.
Int Pass ((5*%e^7*x-%e^7)*%e^(5*x)) ((5*%e^7*x-%e^7)*%e^(5*x))/25+c x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[\frac{e^{5\cdot x+7}}{5}+\frac{\left(5\cdot e^7\cdot x-e^7\right) \cdot e^{5\cdot x}}{5}\] In fact, the derivative of your answer, with respect to \(x\) is: \[5\cdot e^{5\cdot x+7}+5\cdot \left(5\cdot e^7\cdot x-e^7\right) \cdot e^{5\cdot x}\] so you must have done something wrong!
ATInt_generic.
Int Pass ((5*%e^7*x-%e^7)*%e^(5*x)) ((5*%e^7*x-%e^7)*%e^(5*x))/25+c [x,x*%e^(5*x+7)] 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[x\cdot e^{5\cdot x+7}\] In fact, the derivative of your answer, with respect to \(x\) is: \[5\cdot e^{5\cdot x+7}+5\cdot \left(5\cdot e^7\cdot x-e^7\right) \cdot e^{5\cdot x}\] so you must have done something wrong!
ATInt_generic.
Inverse hyperbolic integrals
Int Pass log(x-3)/6-log(x+3)/6+c log(x-3)/6-log(x+3)/6 x 1 1 ATInt_true_equiv.
Int Pass asinh(x) ln(x+sqrt(x^2+1)) x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass asinh(x)+c ln(x+sqrt(x^2+1)) x 1 1 ATInt_true.
Int Pass -acoth(x/3)/3 log(x-3)/6-log(x+3)/6 x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass -acoth(x/3)/3 log(x-3)/6-log(x+3)/6 [x, NOCONST] 1 1 ATInt_true.
Int Pass -acoth(x/3)/3+c log(x-3)/6-log(x+3)/6 x 1 1 ATInt_true.
Int Pass -acoth(x/3)/3+c log(abs(x-3))/6-log(abs(x+3))/6 x 1 1 ATInt_true.
Int Pass log(x-a)/(2*a)-log(x+a)/(2*a)+c log(x-a)/(2*a)-log(x+a)/(2*a) x 1 1 ATInt_true_equiv.
Int Pass -acoth(x/a)/a+c log(x-a)/(2*a)-log(x+a)/(2*a) x 1 1 ATInt_true.
Int Pass -acoth(x/a)/a+c log(abs(x-a))/(2*a)-log(abs(x+a))/(2*a) x 1 1 ATInt_true.
Int Pass log(x-a)/(2*a)-log(x+a)/(2*a)+c log(abs(x-a))/(2*a)-log(abs(x+a))/(2*a) x 0 0
The formal derivative of your answer does equal the expression that you were asked to integrate. However, your answer differs from the correct answer in a significant way, that is to say not just, e.g., a constant of integration. Your teacher may expect you to use the result \(\int\frac{1}{x} dx = \log(|x|)+c\), rather than \(\int\frac{1}{x} dx = \log(x)+c\). Please ask your teacher about this.
ATInt_EqFormalDiff. ATInt_logabs.
Int Pass log(x-3)/6-log(x+3)/6+c -acoth(x/3)/3 x 1 1 ATInt_true.
Int Pass log(abs(x-3))/6-log(abs(x+3))/6+c -acoth(x/3)/3 x 1 1 ATInt_true.
Int Pass log(x-3)/6-log(x+3)/6 -acoth(x/3)/3 x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass atan(2*x-3)+c atan(2*x-3) x 1 1 ATInt_true.
Int Pass atan((x-2)/(x-1))+c atan(2*x-3) x 1 1 ATInt_true.
Int Pass atan((x-2)/(x-1)) atan(2*x-3) x 0 0
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
Int Pass atan((x-1)/(x-2)) atan(2*x-3) x 0 0
The derivative of your answer should be equal to the expression that you were asked to integrate, that was: \[\frac{2}{{\left(2\cdot x-3\right)}^2+1}\] In fact, the derivative of your answer, with respect to \(x\) is: \[\frac{\frac{1}{x-2}-\frac{x-1}{{\left(x-2\right)}^2}}{\frac{{\left( x-1\right)}^2}{{\left(x-2\right)}^2}+1}\] so you must have done something wrong!
ATInt_generic.
Stoutemyer (currently fails)
Int Pass 2/3*sqrt(3)*(atan(sin(x)/(sqrt(3)*(cos(x)+1)))-(atan(sin(x)/(cos(x)+1))))+x/sqrt(3) 2*atan(sin(x)/(sqrt(3)*(cos(x)+1)))/sqrt(3) x 0 -3
You need to add a constant of integration, otherwise this appears to be correct. Well done.
ATInt_const.
GT Expected failure 1/0 1 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATGT_STACKERROR_SAns.
GT Expected failure 1 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATGT_STACKERROR_TAns.
GT Pass 1 1 0 0 ATGT_false.
GT Pass 2 1 1 1 ATGT_true.
GT Pass 1 2.1 0 0 ATGT_false.
GT Pass pi 3 1 1 ATGT_true.
GT Pass pi+2 5 1 1 ATGT_true.
Infinity
GT Pass -inf 0 0 0 Not number
GT Pass inf 0 0 0 Not number
GTE Expected failure 1/0 1 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATGTE_STACKERROR_SAns.
GTE Expected failure 1 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATGTE_STACKERROR_TAns.
GTE Pass 1 1 1 1 ATGTE_true.
GTE Pass 2 1 1 1 ATGTE_true.
GTE Pass 1 2.1 0 0 ATGTE_false.
GTE Pass pi 3 1 1 ATGTE_true.
GTE Pass pi+2 5 1 1 ATGTE_true.
Basic tests
NumRelative Expected failure 1/0 0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATNumRelative_STACKERROR_SAns.
NumRelative Expected failure 0 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATNumRelative_STACKERROR_TAns.
NumRelative Expected failure 0 0 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATNumRelative_STACKERROR_Opt.
NumRelative Expected failure 0 (x The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty teacher answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty teacher answer, probably a CAS validation problem when authoring the question.
ATNumRelativeTEST_FAILED-Empty TA.
NumRelative Expected failure 1.5 1.5 x 0 -1
The numerical tolerance for ATNumerical should be a floating point number, but is not. This is an internal error with the test. Please ask your teacher about this.
ATNumerical_STACKERROR_tol.
NumRelative Pass 1 0 (x 0 0
NumRelative Pass x=1.5 1.5 0 0
Your answer should be a floating point number, but is not.
ATNumerical_SA_not_number.
NumRelative Pass 1.5 x=1.5 0 0
The value supplied for the teacher's answer should be a floating point number, but is not. This is an internal error with the test. Please ask your teacher about this.
ATNumerical_SB_not_number.
No option, so 5%
NumRelative Pass 1.1 1 0 0
NumRelative Pass 1.05 1 1 1
NumRelative Pass 0.95 1 1 1
NumRelative Pass 0.949 1 0 0
NumRelative Pass 1.05e33 1e33 1 1
NumRelative Pass 1.06e33 1e33 0 0
NumRelative Pass 0.95e33 1e33 1 1
NumRelative Pass 0.949e33 1e33 0 0
NumRelative Pass 1.05e-33 1e-33 1 1
NumRelative Pass 1.06e-33 1e-33 0 0
NumRelative Pass 0.95e-33 1e-33 1 1
NumRelative Pass 0.949e-33 1e-33 0 0
Remove display dp etc.
NumRelative Pass 1 displaydp(1.05,2) 0.1 1 1
NumRelative Pass 1000 displaysci(1.05,2,3) 0.1 1 1
Options passed
NumRelative Pass 1.05 1 0.1 1 1
NumRelative Pass 1.05 3 0.1 0 0
NumRelative Pass 3.14 pi 0.001 1 1
Infinity
NumRelative Pass inf 0 0 0
Your answer should be a floating point number, but is not.
ATNumerical_SA_not_number.
Lists
NumRelative Pass 1 [1,2] 0 0
Your answer should be a list, but is not. Note that the syntax to enter a list is to enclose the comma separated values with square brackets.
ATNumerical_SA_not_list.
NumRelative Pass [1,2] [1,2,3] 0 0
Your list should have \(3\) elements, but it actually has \(2\).
ATNumerical_wronglen.
NumRelative Pass [1,2] [1,2] 1 1
NumRelative Pass [3.141,1.414] [pi,sqrt(2)] 1 1
NumRelative Pass [3,1.414] [pi,sqrt(2)] 0.01 0 0
The entries underlined in red below are those that are incorrect. \[\left[ {\color{red}{\underline{3.0}}} , 1.414 \right] \]
ATNumerical_wrongentries SA/TA=[3.0].
NumRelative Pass [3,1.414] {pi,sqrt(2)} 0.01 0 0
Your answer should be a set, but is not. Note that the syntax to enter a set is to enclose the comma separated values with curly brackets.
ATNumerical_SA_not_set.
NumRelative Pass {1.414,3.1} {significantfigures(pi,6),sqrt(2)} 0.01 0 0
The entries underlined in red below are those that are incorrect. \[\left \{{\color{red}{\underline{3.1}}} \right \}\]
ATNumerical_wrongentries: TA/SA=[3.14159], SA/TA=[3.1].
NumRelative Pass {1.414,3.1} {pi,sqrt(2)} 0.1 1 1
Basic tests
NumAbsolute Expected failure 1/0 0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATNumAbsolute_STACKERROR_SAns.
NumAbsolute Expected failure 0 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATNumAbsolute_STACKERROR_TAns.
NumAbsolute Expected failure 0 0 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATNumAbsolute_STACKERROR_Opt.
NumAbsolute Expected failure 0 (x The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty teacher answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty teacher answer, probably a CAS validation problem when authoring the question.
ATNumAbsoluteTEST_FAILED-Empty TA.
NumAbsolute Pass 1 0 (x 0 0
No option, so 5%
NumAbsolute Pass 1.1 1 0 0
NumAbsolute Pass 1.05 1 1 1
Options passed
NumAbsolute Pass 1.05 1 0.1 1 1
NumAbsolute Pass 1.05 3 0.1 0 0
NumAbsolute Pass 3.14 pi 0.001 0 0
NumAbsolute Pass 1.41e-2 1.41e-2 0.0001 1 1
NumAbsolute Pass 0.0141 1.41e-2 0.0001 1 1
NumAbsolute Pass 0.00141 0.00141 0.0001 1 1
NumAbsolute Pass 0.00141 1.41*10^-3 0.0001 1 1
NumAbsolute Pass 1.41*10^-3 1.41*10^-3 0.0001 1 1
NumAbsolute Pass [3.141,1.414] [pi,sqrt(2)] 0.01 1 1
NumAbsolute Pass [3,1.414] [pi,sqrt(2)] 0.01 0 0
The entries underlined in red below are those that are incorrect. \[\left[ {\color{red}{\underline{3.0}}} , 1.414 \right] \]
ATNumerical_wrongentries SA/TA=[3.0].
NumAbsolute Pass [3,1.414] {pi,sqrt(2)} 0.01 0 0
Your answer should be a set, but is not. Note that the syntax to enter a set is to enclose the comma separated values with curly brackets.
ATNumerical_SA_not_set.
NumAbsolute Pass {1.414,3.1} {significantfigures(pi,6),sqrt(2)} 0.01 0 0
The entries underlined in red below are those that are incorrect. \[\left \{{\color{red}{\underline{3.1}}} \right \}\]
ATNumerical_wrongentries: TA/SA=[3.14159], SA/TA=[3.1].
NumAbsolute Pass {1,1.414,3.1,2} {1,2,pi,sqrt(2)} 0.1 1 1
Basic tests
NumSigFigs Expected failure 3.141 3.1415927 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Missing option when executing the test.
STACKERROR_OPTION.
NumSigFigs Expected failure 1/0 3 3 0 -1 ATNumSigFigs_STACKERROR_SAns.
NumSigFigs Expected failure 0 1/0 3 0 -1 ATNumSigFigs_STACKERROR_TAns.
NumSigFigs Expected failure 0 0 1/0 0 -1 ATNumSigFigs_STACKERROR_Opt.
NumSigFigs Expected failure 0 1 ( TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Option field is invalid. You have a missing right bracket ) in the expression: (.
STACKERROR_OPTION.
NumSigFigs Expected failure ( 1 1 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question.
ATNumSigFigsTEST_FAILED-Empty SA.
NumSigFigs Expected failure 1 3 pi 0 -1
The answer test failed to execute correctly: please alert your teacher.
ATNumSigFigs_STACKERROR_not_integer.
NumSigFigs Expected failure 1 3 [3,x] 0 -1
The answer test failed to execute correctly: please alert your teacher.
ATNumSigFigs_STACKERROR_not_integer.
NumSigFigs Expected failure 1 3 [1,2,3] 0 -1
The answer test failed to execute correctly: please alert your teacher.
ATNumSigFigs_STACKERROR_list_wrong_length.
NumSigFigs Expected failure 1 3 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Missing option when executing the test.
STACKERROR_OPTION.
NumSigFigs Pass pi pi 4 0 0
Your answer should be a decimal number, but is not!
ATNumSigFigs_NotDecimal.
Edge cases
NumSigFigs Pass 0 0 2 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0 0 1 1 1
NumSigFigs Pass 0.0 0 1 1 1
NumSigFigs Pass 0.0 0 2 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0 0.0 2 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.0 0.0 2 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.00 0.00 2 1 1
Large numbers
NumSigFigs Pass 5.4e21 5.3e21 2 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 5.3e21 5.3e21 2 1 1
NumSigFigs Pass 5.3e22 5.3e22 2 1 1
NumSigFigs Pass 5.3e20 5.3e22 2 0 0 ATNumSigFigs_VeryInaccurate.
NumSigFigs Pass 6.02214086e23 6.02214086e23 9 1 1
NumSigFigs Pass 6.0221409e23 6.02214086e23 9 0 0
Your answer contains the wrong number of significant digits. The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WrongDigits. ATNumSigFigs_Inaccurate.
NumSigFigs Pass 6.02214087e23 6.02214086e23 9 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 6.02214085e23 6.02214086e23 9 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 5.3910632e-44 5.3910632e-44 8 1 1
NumSigFigs Pass 5.391063e-44 5.3910632e-44 8 0 0
Your answer contains the wrong number of significant digits. The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WrongDigits. ATNumSigFigs_Inaccurate.
NumSigFigs Pass 5.3910631e-44 5.3910632e-44 8 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 5.3910633e-44 5.3910632e-44 8 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 1.61622938e-35 1.61622938e-35 9 1 1
NumSigFigs Pass 1.6162294e-35 1.61622938e-35 9 0 0
Your answer contains the wrong number of significant digits. The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WrongDigits. ATNumSigFigs_Inaccurate.
NumSigFigs Pass 1.61622939e-35 1.61622938e-35 9 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 1.61622937e-35 1.61622938e-35 9 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 1.2345e82 1.2345e82 5 1 1
NumSigFigs Pass 1.2346e82 1.2345e82 5 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 1.2344e82 1.2345e82 5 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
No trailing zeros.
NumSigFigs Pass 1.234 4 1 0 0
Your answer contains the wrong number of significant digits. The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WrongDigits. ATNumSigFigs_Inaccurate.
NumSigFigs Pass 3.141 3.1415927 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 3.141 3.1415927 4 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 3.146 3.1415927 4 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 3.147 3.1415927 4 0 0 ATNumSigFigs_VeryInaccurate.
NumSigFigs Pass 3.142 3.1415927 4 1 1
NumSigFigs Pass 3.142 pi 4 1 1
NumSigFigs Pass 3141 3.1415927 4 0 0 ATNumSigFigs_VeryInaccurate.
NumSigFigs Pass 0.00123 0.001234567 3 1 1
NumSigFigs Pass 1.23e-3 0.001234567 3 1 1
NumSigFigs Pass 138*10^-3 138*10^-3 3 1 1
NumSigFigs Pass -138*10^-3 -138*10^-3 3 1 1
NumSigFigs Pass 138*10^-3 -138*10^-3 3 0 0
Your answer has the wrong algebraic sign.
ATNumSigFigs_WrongSign.
NumSigFigs Pass 1.38*10^-1 138*10^-3 3 1 1
NumSigFigs Pass 1.24e-3 0.001234567 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 1.235e-3 0.001234567 4 1 1
NumSigFigs Pass 1000 999 2 1 1 ATNumSigFigs_WithinRange.
NumSigFigs Pass 1E3 999 2 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass -100 -149 1 1 1
NumSigFigs Pass -0.05 -0.0499 1 1 1
NumSigFigs Pass -(0.05) -0.0499 1 1 1
NumSigFigs Pass 1170 1174.34 3 1 1
NumSigFigs Pass 61300 61250 3 1 1
Previous tricky case
NumSigFigs Pass 0.1667 0.1667 4 1 1
NumSigFigs Pass 0.1666 0.1667 4 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.1663 0.1667 4 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.1662 0.1667 4 0 0 ATNumSigFigs_VeryInaccurate.
NumSigFigs Pass 0.166 0.1667 4 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATNumSigFigs_VeryInaccurate.
NumSigFigs Pass 0.16667 0.1667 4 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
Negative numbers
NumSigFigs Pass -3.141 -3.1415927 4 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass -3.141 -3.1415927 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass -3.141 -3.1415927 4 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass -3.142 -3.1415927 4 1 1
NumSigFigs Pass 3.142 -3.1415927 4 0 0
Your answer has the wrong algebraic sign.
ATNumSigFigs_WrongSign.
NumSigFigs Pass -3.142 3.1415927 4 0 0
Your answer has the wrong algebraic sign.
ATNumSigFigs_WrongSign.
NumSigFigs Pass -3.149 3.1415927 4 0 0
Your answer has the wrong algebraic sign.
ATNumSigFigs_WrongSign. ATNumSigFigs_VeryInaccurate.
NumSigFigs Pass 2.15 75701719/35227192 3 1 1
Round teacher answer
NumSigFigs Pass 0.0499 0.04985 3 1 1
NumSigFigs Pass 0.0498 0.04985 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.0498 0.04975 3 1 1
NumSigFigs Pass 0.0497 0.04975 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.0499 0.0498 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
Final zeros after the decimal are significant.
NumSigFigs Pass 1.5 1.500 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 1.50 1.500 3 1 1
NumSigFigs Pass 1.500 1.500 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 245.0 245 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
Too few digits
NumSigFigs Pass 180 178.35 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WithinRange. ATNumSigFigs_Inaccurate.
NumSigFigs Pass 33 33.1558 3 0 0
Your answer contains the wrong number of significant digits. The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WrongDigits. ATNumSigFigs_Inaccurate.
Mixed options
NumSigFigs Pass 3.142 3.1415927 [4,3] 1 1
NumSigFigs Pass 3.143 3.1415927 [4,3] 1 1
NumSigFigs Pass 3.150 3.1415927 [4,3] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 3.211 3.1415927 [4,3] 0 0 ATNumSigFigs_VeryInaccurate.
NumSigFigs Pass 3.1416 3.1415927 [4,3] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.1666 0.1667 [4,3] 1 1
NumSigFigs Pass 180 178.35 [3,1] 1 1 ATNumSigFigs_WithinRange.
NumSigFigs Pass 33 33.1558 [3,1] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 1.500 1.5 [3,1] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 245.0 245 [3,1] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 12345.7 12345.654321 [6,6] 1 1
NumSigFigs Pass 12345.7 12345.654321 [6,3] 1 1
NumSigFigs Pass 12300.0 12345.654321 [6,3] 1 1
NumSigFigs Pass 12400.0 12345.654321 [6,3] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 13500.0 12345.654321 [6,3] 0 0 ATNumSigFigs_VeryInaccurate.
NumSigFigs Pass 12000.0 12345.654321 [6,2] 1 1
NumSigFigs Pass 13000.0 12345.654321 [6,2] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 11000.0 12345.654321 [6,2] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
Zero option and trailing zeros
NumSigFigs Pass 0.0010 0 [1,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.0010 0 [2,0] 1 1
NumSigFigs Pass 0.0010 0 [3,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.001 0 [1,0] 1 1
NumSigFigs Pass 0.001 0 [2,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.00100 null [2,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.00100 null [3,0] 1 1
NumSigFigs Pass 0.00100 null [4,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 5.00 null [2,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 5.00 null [3,0] 1 1
NumSigFigs Pass 5.00 null [4,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 100 0 [1,0] 1 1
NumSigFigs Pass 100 0 [2,0] 1 1 ATNumSigFigs_WithinRange.
NumSigFigs Pass 100 0 [3,0] 1 1 ATNumSigFigs_WithinRange.
NumSigFigs Pass 100 0 [4,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 10.0 0 [2,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 10.0 0 [3,0] 1 1
NumSigFigs Pass 10.0 0 [4,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0 0 [1,0] 1 1
NumSigFigs Pass 0 0 [2,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.00 0 [1,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.00 0 [2,0] 1 1
NumSigFigs Pass 0.00 0 [3,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 0.00 0 [4,0] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
Condone too many sfs.
NumSigFigs Pass 8.250 8.250 [4,-1] 1 1
NumSigFigs Pass 8.25 8.250 [4,-1] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits.
NumSigFigs Pass 8.250000 8.250 [4,-1] 1 1
NumSigFigs Pass 8.250434 8.250 [4,-1] 1 1
NumSigFigs Pass 82.4 82 [2,-1] 1 1
NumSigFigs Pass 82.5 82 [2,-1] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 83 82 [2,-1] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
1/7 = 0.142857142857...
NumSigFigs Pass 0.1430 1/7 [4,-1] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.1429 1/7 [4,-1] 1 1
NumSigFigs Pass 0.1428 1/7 [4,-1] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.143 1/7 [4,-1] 0 0
Your answer contains the wrong number of significant digits. The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WrongDigits. ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.14284 1/7 [4,-1] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.14285 1/7 [4,-1] 1 1
NumSigFigs Pass 0.14286 1/7 [4,-1] 1 1
NumSigFigs Pass 0.14291 1/7 [4,-1] 1 1
NumSigFigs Pass 0.14294 1/7 [4,-1] 1 1
NumSigFigs Pass 0.14295 1/7 [4,-1] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 0.142 1/7 [2,-1] 1 1
NumSigFigs Pass 0.14290907676 1/7 [2,-1] 1 1
NumSigFigs Pass 0.143 1/7 [2,-1] 1 1
NumSigFigs Pass 0.1433333 1/7 [2,-1] 1 1
NumSigFigs Pass 0.144 1/7 [2,-1] 1 1
NumSigFigs Pass 0.145 1/7 [2,-1] 1 1
NumSigFigs Pass 0.146 1/7 [2,-1] 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
Logarithms, numbers and surds
NumSigFigs Pass 1.279 ev(lg(19),lg=logbasesimp) 4 1 1
NumSigFigs Pass 3.14 pi 3 1 1
NumSigFigs Pass 3.15 pi 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate.
NumSigFigs Pass 1.73205 sqrt(3) 6 1 1
No support for matrices!
NumSigFigs Expected failure matrix([0.33,1],[1,1]) matrix([0.333,1],[1,1]) 2 0 -1
Your answer should be a decimal number, but is not!
ATNumSigFigs_NotDecimal.
NumSigFigs Expected failure 3.1415 matrix([0.333,1],[1,1]) 2 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. sigfigsfun(x,n,d) requires a real number, or a list of real numbers, as a first argument. Received: matrix([0.333,1],[1,1])
TEST_FAILED
Teacher uses dispsf
NumSigFigs Pass 1.50 dispsf(1.500,3) 3 1 1
NumSigFigs Pass 1.50 dispdp(1.500,3) 3 1 1
Basic tests
NumDecPlaces Expected failure 1/0 3 2 0 -1 ATNumDecPlaces_STACKERROR_SAns.
NumDecPlaces Expected failure 0.1 1/0 2 0 -1 ATNumDecPlaces_STACKERROR_TAns.
NumDecPlaces Expected failure 0.1 0 1/0 0 -1 ATNumDecPlaces_STACKERROR_Opt.
NumDecPlaces Expected failure 0.1 1 x 0 -1
For ATNumDecPlaces the test option must be a positive integer, in fact "\(x\)" was received.
ATNumDecPlaces_OptNotInt.
NumDecPlaces Expected failure 0.1 1 -1 0 -1
For ATNumDecPlaces the test option must be a positive integer, in fact "\(-1\)" was received.
ATNumDecPlaces_OptNotInt.
NumDecPlaces Expected failure 0.1 1 0 0 -1
For ATNumDecPlaces the test option must be a positive integer, in fact "\(0\)" was received.
ATNumDecPlaces_OptNotInt.
NumDecPlaces Expected failure 0.1 1 ( TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Option field is invalid. You have a missing right bracket ) in the expression: (.
STACKERROR_OPTION.
NumDecPlaces Expected failure ( 1 1 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question.
ATNumDecPlacesTEST_FAILED-Empty SA.
Student's answer not a floating point number
NumDecPlaces Pass x 3.143 2 0 0
Your answer must be a floating point number, but is not.
ATNumDecPlaces_SA_Not_num.
NumDecPlaces Pass pi 3.000 3 0 0
Your answer must be a floating point number, but is not.
ATNumDecPlaces_SA_Not_num.
Right number of places
NumDecPlaces Pass 3.14 3.143 2 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces Pass 3.14 3.14 2 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces Pass 3.140 3.140 3 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces Pass 3141.5972 3141.5972 4 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces Pass 4.14 3.14 2 0 0 ATNumDecPlaces_Correct. ATNumDecPlaces_Not_equiv.
NumDecPlaces Pass 3.1416 pi 4 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces Pass -7.3 -7.3 1 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
Wrong number of places
NumDecPlaces Pass 3.14 3.143 1 0 0
Your answer has been given to the wrong number of decimal places.
ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Equiv.
NumDecPlaces Pass 3.14 3.143 1 0 0
Your answer has been given to the wrong number of decimal places.
ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Equiv.
NumDecPlaces Pass 3.14 3.140 3 0 0
Your answer has been given to the wrong number of decimal places.
ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Equiv.
NumDecPlaces Pass 7.000 7 4 0 0
Your answer has been given to the wrong number of decimal places.
ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Equiv.
NumDecPlaces Pass 7.0000 7 4 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
Both wrong DPs and inaccurate.
NumDecPlaces Pass 8.0000 7 3 0 0
Your answer has been given to the wrong number of decimal places.
ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Not_equiv.
Teacher needs to round their answer.
NumDecPlaces Pass 4.000 3.99999 3 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
Teacher uses displaydp
NumDecPlaces Pass 0.10 displaydp(0.1,2) 2 1 1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
Basic tests
NumDecPlacesWrong Expected failure 1/0 3 2 0 -1 ATNumDecPlacesWrong_STACKERROR_SAns.
NumDecPlacesWrong Expected failure 0.1 1/0 2 0 -1 ATNumDecPlacesWrong_STACKERROR_TAns.
NumDecPlacesWrong Expected failure 0.1 0 1/0 0 -1 ATNumDecPlacesWrong_STACKERROR_Opt.
NumDecPlacesWrong Expected failure 0.1 0 x 0 -1
For ATNumDecPlacesWrong the test option must be a positive integer, in fact "\(x\)" was received.
ATNumDecPlacesWrong_OptNotInt.
NumDecPlacesWrong Pass x^2 1234 4 0 0
Your answer must be a floating point number, but is not.
ATNumDecPlacesWrong_SA_Not_num.
NumDecPlacesWrong Pass 1234.5 x^2 4 0 0 ATNumDecPlacesWrong_Tans_Not_Num.
NumDecPlacesWrong Pass 3.141 31.41 4 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 3.141 31.14 4 0 0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong Pass pi 31.14 4 0 0
Your answer must be a floating point number, but is not.
ATNumDecPlacesWrong_SA_Not_num.
NumDecPlacesWrong Pass 0.1234 1234 4 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 0.1235 1234 4 0 0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong Pass 0.0001234 1234 4 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 0.0001235 1234 4 0 0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong Pass 0.1233 1234 3 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 0.1243 1234 3 0 0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong Pass 0.1230 1239 3 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 0.1240 1239 3 0 0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong Pass 1230 1239 3 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 2230 1239 3 0 0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong Pass 0.100 1.00 3 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 0.1000 1.00 3 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 0.1001 1.001 3 1 1 ATNumDecPlacesWrong_Correct.
Condone lack of trailing zeros
NumDecPlacesWrong Pass 0.100 1.0 4 1 1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong Pass 1 1.00 4 1 1 ATNumDecPlacesWrong_Correct.
Teacher uses displaydp
NumDecPlacesWrong Pass 0.101 displaydp(101,3) 3 1 1 ATNumDecPlacesWrong_Correct.
Basic tests
SigFigsStrict Expected failure 3.141 null TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Missing option when executing the test.
STACKERROR_OPTION.
SigFigsStrict Expected failure 3.141 null x^2 0 -1 STACKERROR_OPTION.
SigFigsStrict Expected failure 3.141 null -2 0 -1 STACKERROR_OPTION.
SigFigsStrict Expected failure 3.141 null 0 0 -1 STACKERROR_OPTION.
SigFigsStrict Pass 0.0010 null 1 0 0
SigFigsStrict Pass 0.0010 null 2 1 1
SigFigsStrict Pass 0.0010 null 3 0 0
SigFigsStrict Pass 0.00100 null 2 0 0
SigFigsStrict Pass 0.00100 null 3 1 1
SigFigsStrict Pass 0.00100 null 4 0 0
SigFigsStrict Pass 0.001 null 1 1 1
SigFigsStrict Pass 0.001 null 2 0 0
SigFigsStrict Pass 100 null 1 1 1
SigFigsStrict Pass 100 null 2 0 0 ATSigFigsStrict_WithinRange.
SigFigsStrict Pass 100 null 3 0 0 ATSigFigsStrict_WithinRange.
SigFigsStrict Pass 100 null 4 0 0
SigFigsStrict Pass 100. null 1 0 0
SigFigsStrict Pass 100. null 2 0 0
SigFigsStrict Pass 100. null 3 1 1
SigFigsStrict Pass 100. null 4 0 0
SigFigsStrict Pass 123. null 1 0 0
SigFigsStrict Pass 123. null 2 0 0
SigFigsStrict Pass 123. null 3 1 1
SigFigsStrict Pass 123. null 4 0 0
SigFigsStrict Pass 1.00e2 null 1 0 0
SigFigsStrict Pass 1.00e2 null 2 0 0
SigFigsStrict Pass 1.00e2 null 3 1 1
SigFigsStrict Pass 1.00e2 null 4 0 0
SigFigsStrict Pass 10.0 null 2 0 0
SigFigsStrict Pass 10.0 null 3 1 1
SigFigsStrict Pass 10.0 null 4 0 0
SigFigsStrict Pass 0 null 1 1 1
SigFigsStrict Pass 0 null 2 0 0
SigFigsStrict Pass 0.0 null 1 1 1
SigFigsStrict Pass 0.0 null 2 0 0
SigFigsStrict Pass .0 null 1 1 1
SigFigsStrict Pass .0 null 2 0 0
SigFigsStrict Pass .001030 null 4 1 1
SigFigsStrict Pass 0.00 null 1 0 0
SigFigsStrict Pass 0.00 null 2 1 1
SigFigsStrict Pass 0.00 null 3 0 0
SigFigsStrict Pass 25.00e1 null 1 0 0
SigFigsStrict Pass 25.00e1 null 3 0 0
SigFigsStrict Pass 25.00e1 null 4 1 1
SigFigsStrict Pass 25.00e1 null 5 0 0
SigFigsStrict Pass 15.1 15.1 3 1 1
SigFigsStrict Pass 15.10 15.1 3 0 0
SigFigsStrict Pass 15.100 15.1 3 0 0
Units are ignored
SigFigsStrict Pass 9.81*m/s^2 null 3 1 1
Units Expected failure 1/0 1 2 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATUnits_STACKERROR_SAns.
Units Expected failure 1 1/0 2 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATUnits_STACKERROR_TAns.
Units Expected failure 1 1 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATUnits_STACKERROR_Opt.
Units Expected failure x-1)^2 (x-1)^2 2 The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question. 0 -1
The answer test failed to execute correctly: please alert your teacher. Attempted to execute an answer test with an empty student answer, probably a CAS validation problem when authoring the question.
ATUnitsTEST_FAILED-Empty SA.
Units Expected failure 12.3*m*s^(-1) 3*m [3,x] 0 -1
The answer test failed to execute correctly: please alert your teacher.
ATNumSigFigs_STACKERROR_not_integer.
Units Expected failure 3*m*s^(-1) 3*m [1,2,3] 0 -1
The answer test failed to execute correctly: please alert your teacher.
ATNumSigFigs_STACKERROR_list_wrong_length.
Units Expected failure 12.3*m*s^(-1) {12.3*m*s^(-1)} 3 0 -1
The answer test failed to execute correctly: please alert your teacher.
ATUnits_TA_not_expression.
Units Pass x=12.3*m*s^(-1) 12.3*m*s^(-1) 3 0 0
Your answer needs to be a number together with units. Do not use sets, lists, equations or matrices.
ATUnits_SA_not_expression.
Missing units
Units Pass 12.3 12.3*m 3 0 0
Your answer must have units.
ATUnits_SA_no_units.
Units Pass 12 12.3*m 3 0 0
Your answer must have units.
ATUnits_SA_no_units.
Units Pass 1/2 12.3*m 3 0 0
Your answer must have units.
ATUnits_SA_no_units.
Units Pass e^(1/2) 12.3*m 3 0 0
Your answer must have units.
ATUnits_SA_no_units.
Units Expected failure 9.81*m 12.3 3 0 -1
The answer test failed to execute correctly: please alert your teacher.
ATUnits_SB_no_units.
Only units
Units Pass m/s 12.3*m/s 3 0 0
Your answer needs to be a number together with units. Your answer only has units.
ATUnits_SA_only_units.
Units Pass m 12.3*m/s 3 0 0
Your answer needs to be a number together with units. Your answer only has units.
ATUnits_SA_only_units.
Bad units
Units Pass 9.81+m/s 9.81*m/s 3 0 0
Your answer must have units, and you must use multiplication to attach the units to a value, e.g. 3.2*m/s.
ATUnits_SA_bad_units.
Basic tests
Units Pass 12.3*m/s 12.3*m/s 3 1 1 ATUnits_units_match.
Units Pass 12.4*m/s 12.3*m/s 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate. ATUnits_units_match.
Units Pass 12.4*m/s 12.3*m/s [3,2] 1 1 ATUnits_units_match.
Units Pass 12.45*m/s 12.3*m/s [3,2] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATUnits_units_match.
Units Pass 13.45*m/s 12.3*m/s [3,2] 0 0
Your answer contains the wrong number of significant digits. The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WrongDigits. ATNumSigFigs_Inaccurate. ATUnits_units_match.
Units Pass 7.54E-5*(s*M)^-1 5.625E-5*s^-1 [3,2] 0 0
Your units are incompatible with those used by the teacher.
ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
Units Pass 7.54E-5*(s*M)^-1 stackunits(5.625E-5,1/s) [3,2] 0 0
Your units are incompatible with those used by the teacher.
ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
Units Pass 12*m/s 12.3*m/s 3 0 0
Your answer contains the wrong number of significant digits. The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WrongDigits. ATNumSigFigs_Inaccurate. ATUnits_units_match.
Units Pass -9.81*m/s^2 -9.81*m/s^2 3 1 1 ATUnits_units_match.
Units Pass -9.82*m/s^2 -9.815*m/s^2 3 1 1 ATUnits_units_match.
Units Pass -9.81*m/s^2 -9.815*m/s^2 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate. ATUnits_units_match.
Units Pass -9.81*m*s^(-2) -9.81*m/s^2 3 1 1 ATUnits_units_match.
Units Pass -9.82*m/s^2 -9.81*m/s^2 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate. ATUnits_units_match.
Units Pass -9.81*m*s^(-2) -9.81*m/s^2 3 1 1 ATUnits_units_match.
Units Pass -9.81*m/s/s -9.81*m/s^2 3 1 1 ATUnits_units_match.
Units Pass -9.81*m/s -9.81*m/s^2 3 0 0
Your units are incompatible with those used by the teacher. Please check your units carefully.
ATUnits_incompatible_units. ATUnits_correct_numerical.
Units Pass -9.81*m/s -9.81*m/s^2 3 0 0
Your units are incompatible with those used by the teacher. Please check your units carefully.
ATUnits_incompatible_units. ATUnits_correct_numerical.
Units Pass (-9.81)*m/s^2 -9.81*m/s^2 3 1 1 ATUnits_units_match.
Units Pass 520*amu 520*amu 3 1 1 ATNumSigFigs_WithinRange. ATUnits_units_match.
Units Pass 520*amu 521*amu 3 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_WithinRange. ATNumSigFigs_Inaccurate. ATUnits_units_match.
Missing units
Units Pass (-9.81) -9.81*m/s^2 3 0 0
Your answer must have units.
ATUnits_SA_no_units.
Units Pass 9.81*m/s -9.81*m/s^2 3 0 0
Your answer has the wrong algebraic sign. Your units are incompatible with those used by the teacher.
ATNumSigFigs_WrongSign. ATUnits_incompatible_units.
Units Pass 8.81*m/s -9.81*m/s^2 3 0 0
Your answer has the wrong algebraic sign. Your units are incompatible with those used by the teacher.
ATNumSigFigs_WrongSign. ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
Units Pass 8.1*m/s -9.81*m/s^2 3 0 0
Your answer contains the wrong number of significant digits. Your answer has the wrong algebraic sign. Your units are incompatible with those used by the teacher.
ATNumSigFigs_WrongDigits. ATNumSigFigs_WrongSign. ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
Units Pass m/4 0.25*m 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATUnits_units_match.
Student is too exact
Units Pass pi*s 3.14*s 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATUnits_units_match.
Units Pass sqrt(2)*m 1.41*m 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATUnits_units_match.
Different units
Units Pass 25*g 0.025*kg 2 1 1 ATUnits_compatible_units kg.
Units Pass 26*g 0.025*kg 2 0 0
The accuracy of your answer is not correct. Either you have not rounded correctly, or you have rounded an intermediate answer which propagates an error.
ATNumSigFigs_Inaccurate. ATUnits_compatible_units kg.
Units Pass 100*g 10*kg 2 0 0 ATNumSigFigs_WithinRange. ATNumSigFigs_VeryInaccurate. ATUnits_compatible_units kg.
Units Pass 0.025*g 0.025*kg 2 0 0
Please check your units carefully.
ATUnits_compatible_units kg. ATUnits_correct_numerical.
Units Pass 1000*m 1*km 2 1 1 ATNumSigFigs_WithinRange. ATUnits_compatible_units m.
Units Pass 1*Mg/10^6 1*N*s^2/(km) 1 1 1 ATUnits_compatible_units kg.
Units Pass 1*Mg/10^6 1*kN*ns/(mm*Hz) 1 1 1 ATUnits_compatible_units kg.
Units Pass 3.14*Mg/10^6 %pi*kN*ns/(mm*Hz) 3 1 1 ATUnits_compatible_units kg.
Units Pass 3.141*Mg/10^6 %pi*kN*ns/(mm*Hz) 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATUnits_compatible_units kg.
Units Pass 4.141*Mg/10^6 %pi*kN*ns/(mm*Hz) 3 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATNumSigFigs_VeryInaccurate. ATUnits_compatible_units kg.
Units Pass 400*cc 0.4*l 2 1 1 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
Units Pass 400*cm^3 0.4*l 2 1 1 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
Units Pass 400*ml 0.4*l 2 1 1 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
Units Pass 18*kJ 18000.0*J 2 1 1 ATUnits_compatible_units (kg*m^2)/s^2.
Units Pass 18.1*kJ 18000.0*J 2 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATUnits_compatible_units (kg*m^2)/s^2.
Units Pass 120*kWh 0.12*MWh 2 1 1 ATUnits_compatible_units (kg*m^2)/s^2.
Units Pass 2.0*hh 720000*s 2 1 1 ATUnits_compatible_units s.
Units Pass 723*kVA 0.723*MVA 3 1 1 ATUnits_compatible_units (kg*m^2)/s^3.
Edge case
Units Pass 0*m/s 0*m/s 1 1 1 ATUnits_units_match.
Units Pass 0.0*m/s 0*m/s 1 1 1 ATUnits_units_match.
Units Pass 0*m/s 0.0*m/s 1 1 1 ATUnits_units_match.
Units Pass 0.00*m/s 0.0*m/s 2 1 1 ATUnits_units_match.
Units Pass 0.0*km/s 0.0*m/s 1 1 1 ATUnits_compatible_units m/s.
Units Pass 0.0*m 0.0*m/s 1 0 0
Your units are incompatible with those used by the teacher. Please check your units carefully.
ATUnits_incompatible_units. ATUnits_correct_numerical.
Units Pass 0.0 0.0*m/s 1 0 0
Your answer must have units.
ATUnits_SA_no_units.
Imperial
Units Pass 7*in 7*in 1 1 1 ATUnits_units_match.
Units Pass 6*in 0.5*ft 1 1 1 ATUnits_compatible_units in.
Units Pass 2640*ft 0.5*mi 4 1 1 ATNumSigFigs_WithinRange. ATUnits_compatible_units in.
Units Pass 2650*ft 0.5*mi 4 0 0 ATNumSigFigs_WithinRange. ATNumSigFigs_VeryInaccurate. ATUnits_compatible_units in.
TODO
Units Pass 142.8*C 415.9*K 4 0 -3
Your units are incompatible with those used by the teacher.
ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
Units Pass 520*mamu 520*mamu 3 0 -3
The answer test failed to execute correctly: please alert your teacher.
ATUnits_SB_no_units.
Differences from the Units test only
UnitsStrict Pass 25*g 0.025*kg 2 0 0 ATUnits_compatible_units kg.
UnitsStrict Pass 1*Mg/10^6 1*N*s^2/(km) 1 0 0 ATUnits_compatible_units kg.
UnitsStrict Pass 1*Mg/10^6 1*kN*ns/(mm*Hz) 1 0 0 ATUnits_compatible_units kg.
UnitsStrict Pass 3.14*Mg/10^6 %pi*kN*ns/(mm*Hz) 3 0 0 ATUnits_compatible_units kg.
UnitsStrict Pass 400*cc 0.4*l 2 0 0 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
UnitsStrict Pass 400*cm^3 0.4*l 2 0 0 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
UnitsStrict Pass 400*ml 0.4*l 2 0 0 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
UnitsStrict Pass 400*mL 0.4*l 2 0 0 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
UnitsStrict Pass 142.8*C 415.9*K 4 0 0 ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
We are not *that* strict!
UnitsStrict Pass -9.81*m/s/s -9.81*m/s^2 3 1 1 ATUnits_units_match.
Edge case
UnitsStrict Pass 0*m/s 0*m/s 1 1 1 ATUnits_units_match.
UnitsStrict Pass 0.0*m/s 0*m/s 1 1 1 ATUnits_units_match.
UnitsStrict Pass 0*m/s 0.0*m/s 1 1 1 ATUnits_units_match.
UnitsStrict Pass 0.0*m/s 0.0*m/s 1 1 1 ATUnits_units_match.
UnitsStrict Pass 0.0*km/s 0.0*m/s 1 0 0 ATUnits_compatible_units m/s.
UnitsStrict Pass 0.0*m 0.0*m/s 1 0 0 ATUnits_incompatible_units. ATUnits_correct_numerical.
UnitsStrict Pass 0.0 0.0*m/s 1 0 0
Your answer must have units.
ATUnits_SA_no_units.
UnitsStrict Pass 2.33e-15*kg 2.33e-15*kg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 7.03e-3*ng 7.03e-3*ng [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 2.35e-6*ug 2.35e-6*ug [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 9.83e-10*cg 9.83e-10*cg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 9.73e-21*Gg 9.73e-21*Gg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 7.19e-15*kg 7.19e-15*kg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 8.12e-12*g 8.12e-12*g [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 9.34e-12*g 9.34e-12*g [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 1.07e-21*Gg 1.07e-21*Gg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 1.91e-10*cg 1.91e-10*cg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 5.67e-18*Mg 5.67e-18*Mg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 2.04e-9*mg 2.04e-9*mg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 6.75e-6*ug 6.75e-6*ug [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 6.58e-6*ug 6.58e-6*ug [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 3.58e-9*mg 3.58e-9*mg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 9.99e-15*kg 9.99e-15*kg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 9.8e-9*mg 9.8e-9*mg [3,2] 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATUnits_units_match.
UnitsStrict Pass 9.80e-9*mg 9.8e-9*mg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 9.83e-9*mg 9.8e-9*mg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 9.78e-9*mg 9.8e-9*mg [3,2] 1 1 ATUnits_units_match.
UnitsStrict Pass 36*Kj/mol 36*Kj/mol 2 1 1 ATUnits_units_match.
UnitsStrict Pass -36*Kj/mol -36*Kj/mol 2 1 1 ATUnits_units_match.
UnitsStrict Pass (-36)*Kj/mol -36*Kj/mol 2 1 1 ATUnits_units_match.
UnitsStrict Pass (-36*Kj)/mol -36*Kj/mol 2 1 1 ATUnits_units_match.
UnitsStrict Pass -(36*Kj)/mol -36*Kj/mol 2 1 1 ATUnits_units_match.
UnitsStrict Pass -(36.2*Kj)/mol -36.3*Kj/mol 2 0 0
Your answer contains the wrong number of significant digits.
ATNumSigFigs_WrongDigits. ATUnits_units_match.
UnitsRelative Pass 12.3*m/s 12.3*m/s 0.01 1 1 ATUnits_units_match.
UnitsRelative Pass 12*m/s 12.3*m/s 0.01 0 0 ATUnits_units_match.
UnitsRelative Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.15 1 1 ATUnits_compatible_units kg.
UnitsRelative Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.05 0 0 ATUnits_compatible_units kg.
Edge case
UnitsRelative Pass 0*m/s 0*m/s 0.01 1 1 ATUnits_units_match.
UnitsRelative Pass 0.0*m/s 0*m/s 0.01 1 1 ATUnits_units_match.
UnitsRelative Pass 0*m/s 0.0*m/s 0.01 1 1 ATUnits_units_match.
UnitsRelative Pass 0.0*m/s 0.0*m/s 0.01 1 1 ATUnits_units_match.
UnitsRelative Pass 0.0*km/s 0.0*m/s 0.01 1 1 ATUnits_compatible_units m/s.
UnitsRelative Pass 0.0*m 0.0*m/s 0.01 0 0
Your units are incompatible with those used by the teacher. Please check your units carefully.
ATUnits_incompatible_units. ATUnits_correct_numerical.
UnitsRelative Pass 0.0 0.0*m/s 0.01 0 0
Your answer must have units.
ATUnits_SA_no_units.
UnitsRelative Pass 0.0*kVA 0.0*kVA 0.002 1 1 ATUnits_units_match.
UnitsStrictRelative Pass 12.3*m/s 12.3*m/s 0.01 1 1 ATUnits_units_match.
UnitsStrictRelative Pass 12*m/s 12.3*m/s 0.01 0 0 ATUnits_units_match.
UnitsStrictRelative Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.15 0 0 ATUnits_compatible_units kg.
UnitsStrictRelative Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.05 0 0 ATUnits_compatible_units kg.
Edge case
UnitsStrictRelative Pass 0*m/s 0*m/s 0.01 1 1 ATUnits_units_match.
UnitsStrictRelative Pass 0.0*m/s 0*m/s 0.01 1 1 ATUnits_units_match.
UnitsStrictRelative Pass 0*m/s 0.0*m/s 0.01 1 1 ATUnits_units_match.
UnitsStrictRelative Pass 0.0*m/s 0.0*m/s 0.01 1 1 ATUnits_units_match.
UnitsStrictRelative Pass 0.0*km/s 0.0*m/s 0.01 0 0 ATUnits_compatible_units m/s.
UnitsStrictRelative Pass 0.0*m 0.0*m/s 0.01 0 0 ATUnits_incompatible_units. ATUnits_correct_numerical.
UnitsStrictRelative Pass 0.0 0.0*m/s 0.01 0 0
Your answer must have units.
ATUnits_SA_no_units.
UnitsStrictRelative Pass 0*J 0.0*J 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass -123000*J -123*kJ 5*J 0 0
The units specified for the numerical tolerance must match the units used for the teacher's answer. This is an internal error with the test. Please ask your teacher about this.
ATUnits_SO_wrong_units.
UnitsAbsolute Pass 12.3*m/s 12.3*m/s 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass 12*m/s 12.3*m/s 0.01 0 0 ATUnits_units_match.
UnitsAbsolute Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.15 1 1 ATUnits_compatible_units kg.
The following illustrates that we convert to base units to compare.
UnitsAbsolute Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.1 1 1 ATUnits_compatible_units kg.
UnitsAbsolute Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.09 0 0 ATUnits_compatible_units kg.
Units in the options
UnitsAbsolute Pass -123000*J -123*kJ 5*kJ 1 1 ATUnits_compatible_units (kg*m^2)/s^2.
UnitsAbsolute Pass -123006*J -123*kJ 5*kJ 1 1 ATUnits_compatible_units (kg*m^2)/s^2.
UnitsAbsolute Pass -129006*J -123*kJ 5*kJ 0 0 ATUnits_compatible_units (kg*m^2)/s^2.
UnitsAbsolute Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.1*kN*ns/(mm*Hz) 1 1 ATUnits_compatible_units kg.
UnitsAbsolute Pass 1.1*Mg/10^6 1.2*kN*ns/(mm*Hz) 0.09*kN*ns/(mm*Hz) 0 0 ATUnits_compatible_units kg.
Edge case
UnitsAbsolute Pass 0*m/s 0*m/s 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass 0.0*m/s 0*m/s 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass 0*m/s 0.0*m/s 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass 0.0*m/s 0.0*m/s 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass 0.0*km/s 0.0*m/s 0.01 1 1 ATUnits_compatible_units m/s.
UnitsAbsolute Pass 0.0*m 0.0*m/s 0.01 0 0
Your units are incompatible with those used by the teacher. Please check your units carefully.
ATUnits_incompatible_units. ATUnits_correct_numerical.
UnitsAbsolute Pass 0.0 0.0*m/s 0.01 0 0
Your answer must have units.
ATUnits_SA_no_units.
UnitsAbsolute Pass 1.0*m/s m/s 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass 15/pi*kN/mm^2 15/pi*kN/mm^2 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass (15*kN)/(pi*mm^2) (15*kN)/(pi*mm^2) 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass (15/pi)*(kN/mm^2) (15/pi)*(kN/mm^2) 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass (600*N)/(%pi*mm^2) (600*N)/(%pi*mm^2) 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass (600/pi)*kN/m^2 (600/pi)*kN/m^2 0.01 1 1 ATUnits_units_match.
UnitsAbsolute Pass (600/pi)*kN/mm^2 (600/pi)*kN/mm^2 0.01 1 1 ATUnits_units_match.
String Pass Hello hello 0 0
String Pass hello hello 1 1
String Pass hello heloo 0 0
StringSloppy Pass hello Hello 1 1
StringSloppy Pass hel lo Hello TEST_FAILED 0 0
The answer test failed to execute correctly: please alert your teacher. Illegal spaces found in expression hel_lo.
ATStringSloppy_STACKERROR_SAns.
StringSloppy Pass hel lo Hel*lo TEST_FAILED 0 0
The answer test failed to execute correctly: please alert your teacher. Illegal spaces found in expression hel_lo.
ATStringSloppy_STACKERROR_SAns.
StringSloppy Pass hello heloo 0 0
SRegExp Expected failure 1/0 "hello" TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSRegExp_STACKERROR_SAns.
SRegExp Expected failure "1/0" 1/0 TEST_FAILED 0 -1
The answer test failed to execute correctly: please alert your teacher. Division by zero.
ATSRegExp_STACKERROR_TAns.
SRegExp Expected failure Hello hello 0 -1
The second argument to the SRegExp answer test must be a string. The test failed. Please contact your teacher.
ATSRegExp_SB_not_string.
SRegExp Expected failure Hello "hello" 0 -1
The first argument to the SRegExp answer test must be a string. The test failed. Please contact your teacher.
ATSRegExp_SA_not_string.
SRegExp Pass "aaaaabbb" "(aaa)*b" 1 1 ATSRegExp: ["aaab","aaa"].
SRegExp Pass "aab" "(aaa)*b" 1 1 ATSRegExp: ["b",false].
SRegExp Pass "aaac" "(aaa)*b" 0 0
Anchor pattern to the start and the end of the string
SRegExp Pass "aab" "^[aA]*b$" 1 1 ATSRegExp: ["aab"].
SRegExp Pass "aab" "^(aaa)*b$" 0 0
SRegExp Pass "aAb" "^[aA]*b$" 1 1 ATSRegExp: ["aAb"].
SRegExp Pass " aAb" "^[aA]*b$" 0 0
Case insensitive
SRegExp Pass "caAb" "(?i:a*b)" 1 1 ATSRegExp: ["aAb"].
Options
SRegExp Pass "Alice went to the market" "(Alice|Bob) went to the (bank|market)" 1 1 ATSRegExp: ["Alice went to the market","Alice","market"].
SRegExp Pass "Malice went to the shop" "(Alice|Bob) went to the (bank|market)" 0 0
Whitespace, note test rendering issue, the test string has additional spaces and tabs as does the result
SRegExp Pass "Alice went to the market" "(Alice|Bob)\\s+went\\s+to\\s+the\\s+(bank|market)" 1 1 ATSRegExp: ["Alice went to the market","Alice","market"].
SRegExp Pass "Alice went to themarket" "(Alice|Bob)\\s+went\\s+to\\s+the\\s+(bank|market)" 0 0
Escaping patterns, note the function that does it
SRegExp Pass "x^2.2" "x\\^2\\.2" 1 1 ATSRegExp: ["x^2.2"].
SRegExp Pass "x^2+sin(x)" sconcat(string_to_regex("sin(x)"),"$") 1 1 ATSRegExp: ["sin(x)"].
SRegExp Pass "sin(x)+x^2" sconcat(string_to_regex("sin(x)"),"$") 0 0
LowestTerms Expected failure 1/0 0 0 -1 ATLowestTerms_STACKERROR_SAns.
Mix of floats and rational numbers
LowestTerms Pass 0.5 0 1 1
LowestTerms Pass 0.33 0 1 1
LowestTerms Pass 2/4 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{2}{4} \right] \] Please try again.
ATLowestTerms_entries.
Negative numbers
LowestTerms Pass -1/3 0 1 1
LowestTerms Pass 1/-3 0 1 1
LowestTerms Pass -2/4 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{-2}{4} \right] \] Please try again.
ATLowestTerms_entries.
LowestTerms Pass 2/-4 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{2}{-4} \right] \] Please try again.
ATLowestTerms_entries.
LowestTerms Pass -1/-3 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{-1}{-3} \right] \] Please try again.
ATLowestTerms_entries.
LowestTerms Pass -2/-4 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{-2}{-4} \right] \] Please try again.
ATLowestTerms_entries.
Polynomials
LowestTerms Pass x+1/3 0 1 1
LowestTerms Pass x+2/6 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{2}{6} \right] \] Please try again.
ATLowestTerms_entries.
LowestTerms Pass 2*x/4+2/6 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{2}{6} \right] \] Please try again.
ATLowestTerms_entries.
LowestTerms Pass 2/4*x+2/6 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{2}{4} , \frac{2}{6} \right] \] Please try again.
ATLowestTerms_entries.
LowestTerms Pass x-1/-4 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{-1}{-4} \right] \] Please try again.
ATLowestTerms_entries.
Trig functions
LowestTerms Pass cos(x) 0 1 1
LowestTerms Pass cos(3/6*x) 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{3}{6} \right] \] Please try again.
ATLowestTerms_entries.
Matrices
LowestTerms Pass matrix([1,2/4],[2,3]) 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{2}{4} \right] \] Please try again.
ATLowestTerms_entries.
Equations
LowestTerms Pass x=1/2 0 1 1
LowestTerms Pass 3/9=x 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{3}{9} \right] \] Please try again.
ATLowestTerms_entries.
Use predicate lowesttermsp
LowestTerms Pass x^2/x 0 1 1
LowestTerms Pass (2*x)/(4*t) 0 1 1
LowestTerms Pass (2/4)*(x^2/t) 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{2}{4} \right] \] Please try again.
ATLowestTerms_entries.
LowestTerms Pass x^(2/4) 0 0 0
The following terms in your answer are not in lowest terms. \[\left[ \frac{2}{4} \right] \] Please try again.
ATLowestTerms_entries.
Need to rationalize demoninator
LowestTerms Pass sqrt(3)/3 sqrt(3)/3 1 1
LowestTerms Pass 1/sqrt(3) sqrt(3)/3 0 0
You must clear the following from the denominator of your fraction: \[\left[ \sqrt{3} \right] \]
ATLowestTerms_not_rat.
LowestTerms Pass 1/(1-sqrt(2)) 1/(1-sqrt(2)) 0 0
You must clear the following from the denominator of your fraction: \[\left[ \sqrt{2} \right] \]
ATLowestTerms_not_rat.
LowestTerms Pass 1/(1+i) (1-i)/2 0 0
You must clear the following from the denominator of your fraction: \[\left[ \mathrm{i} \right] \]
ATLowestTerms_not_rat.
LowestTerms Pass 1+2/sqrt(3) (2*sqrt(3)+3)/3 0 0
You must clear the following from the denominator of your fraction: \[\left[ \sqrt{3} \right] \]
ATLowestTerms_not_rat.
LowestTerms Pass 1/(1+1/root(3,2)) sqrt(3)/(sqrt(3)+1) 0 0
You must clear the following from the denominator of your fraction: \[\left[ 3^{\frac{1}{2}} \right] \]
ATLowestTerms_not_rat.
LowestTerms Pass 1/(1+1/root(2,3)) 1/(1+1/root(2,3)) 0 0
You must clear the following from the denominator of your fraction: \[\left[ 2^{\frac{1}{3}} \right] \]
ATLowestTerms_not_rat.
Equiv Pass [] [] 1 1
\[\begin{array}{lll} &\left[ \right] & \cr \end{array}\]
[EMPTYCHAR]
Equiv Pass [x^2=-1] [] 1 1
\[\begin{array}{lll} &x^2=-1& \cr \end{array}\]
[EMPTYCHAR]
Equiv Pass [x=x,all] [] 1 1
\[\begin{array}{lll} &x=x& \cr \color{green}{\Leftrightarrow}&\mathbb{R}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [x=x,true] [] 1 1
\[\begin{array}{lll} &x=x& \cr \color{green}{\Leftrightarrow}&\mathbf{True}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [x=x,false] [] 0 0
\[\begin{array}{lll} &x=x& \cr \color{red}{?}&\mathbf{False}& \cr \end{array}\]
[EMPTYCHAR,QMCHAR]
Equiv Pass [1=1,all] [] 1 1
\[\begin{array}{lll} &1=1& \cr \color{green}{\Leftrightarrow}&\mathbb{R}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [1=1,true] [] 1 1
\[\begin{array}{lll} &1=1& \cr \color{green}{\Leftrightarrow}&\mathbf{True}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [0=0,all] [] 1 1
\[\begin{array}{lll} &0=0& \cr \color{green}{\Leftrightarrow}&\mathbb{R}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [0=0,true] [] 1 1
\[\begin{array}{lll} &0=0& \cr \color{green}{\Leftrightarrow}&\mathbf{True}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [1=2,false] [] 1 1
\[\begin{array}{lll} &1=2& \cr \color{green}{\Leftrightarrow}&\mathbf{False}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [1=2,none] [] 1 1
\[\begin{array}{lll} &1=2& \cr \color{green}{\Leftrightarrow}&\emptyset& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [1=2,{}] [] 1 1
\[\begin{array}{lll} &1=2& \cr \color{green}{\Leftrightarrow}&\left \{ \right \}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [1=2,[]] [] 1 1
\[\begin{array}{lll} &1=2& \cr \color{green}{\Leftrightarrow}&\left[ \right] & \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [x=1,X=1] [] 0 0
\[\begin{array}{lll} &x=1& \cr \color{red}{?}&X=1& \cr \end{array}\]
[EMPTYCHAR,QMCHAR]
Equiv Pass [1/(x^2+1)=1/((x+%i)*(x-%i)),true] [] 1 1
\[\begin{array}{lll} &\frac{1}{x^2+1}=\frac{1}{\left(x+\mathrm{i}\right)\cdot \left(x-\mathrm{i}\right)}& \cr \color{green}{\Leftrightarrow}&\mathbf{True}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [2^2,stackeq(4)] [] 1 1
\[\begin{array}{lll} &2^2& \cr \color{green}{\checkmark}&=4& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK]
Equiv Pass [2^2,stackeq(3)] [] 0 0
\[\begin{array}{lll} &2^2& \cr \color{red}{\Rightarrow}&=3& \cr \end{array}\]
[EMPTYCHAR,IMPLIESCHAR]
Equiv Pass [2^2,4] [] 1 1
\[\begin{array}{lll} &2^2& \cr \color{green}{\Leftrightarrow}&4& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [2^2,3] [] 0 0
\[\begin{array}{lll} &2^2& \cr \color{red}{\Rightarrow}&3& \cr \end{array}\]
[EMPTYCHAR,IMPLIESCHAR]
Equiv Pass [lg(64,4),lg(4^3,4),3*lg(4,4),3] [] 1 1
\[\begin{array}{lll} &\log_{4}\left(64\right)& \cr \color{green}{\Leftrightarrow}&\log_{4}\left(4^3\right)& \cr \color{green}{\Leftrightarrow}&3\cdot \log_{4}\left(4\right)& \cr \color{green}{\Leftrightarrow}&3& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [lg(64,4),stackeq(lg(4^3,4)),stackeq(3*lg(4,4)),stackeq(3)] [] 1 1
\[\begin{array}{lll} &\log_{4}\left(64\right)& \cr \color{green}{\checkmark}&=\log_{4}\left(4^3\right)& \cr \color{green}{\checkmark}&=3\cdot \log_{4}\left(4\right)& \cr \color{green}{\checkmark}&=3& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK]
Equiv Pass [x=1 or x=2,x=1 or 2] [] 0 0
\[\begin{array}{lll} &x=1\,{\mbox{ or }}\, x=2& \cr \color{red}{\mbox{Missing assignments}}&x=1\,{\mbox{ or }}\, 2& \cr \end{array}\]
[EMPTYCHAR,MISSINGVAR]
Equiv Pass [x=1 or x=2,x=1 and x=2] [] 0 0
\[\begin{array}{lll} &x=1\,{\mbox{ or }}\, x=2& \cr \color{red}{\mbox{and/or confusion!}}&\left\{\begin{array}{l}x=1\cr x=2\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR,ANDOR]
Equiv Pass [x=1 and y=2,x=1 or y=2] [] 0 0
\[\begin{array}{lll} &\left\{\begin{array}{l}x=1\cr y=2\cr \end{array}\right.& \cr \color{red}{\mbox{and/or confusion!}}&x=1\,{\mbox{ or }}\, y=2& \cr \end{array}\]
[EMPTYCHAR,ANDOR]
Equiv Pass [a=b,a^2=b^2] [] 0 0
\[\begin{array}{lll} &a=b& \cr \color{red}{\Rightarrow}&a^2=b^2& \cr \end{array}\]
[EMPTYCHAR,IMPLIESCHAR]
Equiv Pass [a=b,sqrt(a)=sqrt(b)] [] 0 0
\[\begin{array}{lll} &a=b& \cr \color{red}{\Leftarrow}&\sqrt{a}=\sqrt{b}& \cr \end{array}\]
[EMPTYCHAR,IMPLIEDCHAR]
Equiv Pass [a^2=b^2,a=b] [] 0 0
\[\begin{array}{lll} &a^2=b^2& \cr \color{red}{\Leftarrow}&a=b& \cr \end{array}\]
[EMPTYCHAR,IMPLIEDCHAR]
Equiv Pass [a^2=b^2,a=b or a=-b] [] 1 1
\[\begin{array}{lll} &a^2=b^2& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=-b& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [a^2=b^2,a= #pm#b,a= b or a=-b] [] 1 1
\[\begin{array}{lll} &a^2=b^2& \cr \color{green}{\Leftrightarrow}&a= \pm b& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=-b& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a=b,abs(a)=abs(b),a=b] [] 0 0
\[\begin{array}{lll} &a=b& \cr \color{red}{\Rightarrow}&\left| a\right| =\left| b\right| & \cr \color{red}{\Leftarrow}&a=b& \cr \end{array}\]
[EMPTYCHAR,IMPLIESCHAR,IMPLIEDCHAR]
Equiv Pass [abs(a)=abs(b),a=b or a=-b] [] 1 1
\[\begin{array}{lll} &\left| a\right| =\left| b\right| & \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=-b& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [abs(a)=abs(b),a^2=b^2] [] 1 1
\[\begin{array}{lll} &\left| a\right| =\left| b\right| & \cr \color{green}{\Leftrightarrow}&a^2=b^2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [x^3=8,x=2] [] 0 0
\[\begin{array}{lll} &x^3=8& \cr \color{red}{\Leftarrow}&x=2& \cr \end{array}\]
[EMPTYCHAR,IMPLIEDCHAR]
Equiv Pass [x^3=8,x=2] [] [assumereal] 1 1
\[\begin{array}{lll}\color{blue}{(\mathbb{R})}&x^3=8& \cr \color{green}{\Leftrightarrow}\, \color{blue}{(\mathbb{R})}&x=2& \cr \end{array}\]
[ASSUMEREALVARS, EQUIVCHARREAL]
Equiv Pass [abs(x-1/2)+abs(x+1/2)=2,abs(x)=1] [] 1 1
\[\begin{array}{lll} &\left| x-\frac{1}{2}\right| +\left| x+\frac{1}{2}\right| =2& \cr \color{green}{\Leftrightarrow}&\left| x\right| =1& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [a^2=9 and a>0,a=3] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}a^2=9\cr a > 0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&a=3& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [T=2*pi*sqrt(L/g),T^2=4*pi^2*L/g,g=4*pi^2*L/T^2] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&T=2\cdot \pi\cdot \sqrt{\frac{L}{g}}& \cr \color{green}{\Leftrightarrow}&T^2=\frac{4\cdot \pi^2\cdot L}{g}& \cr \color{green}{\Leftrightarrow}&g=\frac{4\cdot \pi^2\cdot L}{T^2}& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a=b,a^2=b^2] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&a=b& \cr \color{green}{\Leftrightarrow}&a^2=b^2& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [a=b,sqrt(a)=sqrt(b)] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&a=b& \cr \color{green}{\Leftrightarrow}&\sqrt{a}=\sqrt{b}& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [a^2=b^2,a=b] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&a^2=b^2& \cr \color{green}{\Leftrightarrow}&a=b& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [a^2=b^2,a=b or a=-b] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&a^2=b^2& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=-b& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [a=b,abs(a)=abs(b)] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&a=b& \cr \color{green}{\Leftrightarrow}&\left| a\right| =\left| b\right| & \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [abs(a)=abs(b),a=b] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&\left| a\right| =\left| b\right| & \cr \color{green}{\Leftrightarrow}&a=b& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [abs(a)=abs(b),a=-b] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&\left| a\right| =\left| b\right| & \cr \color{green}{\Leftrightarrow}&a=-b& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [abs(a)=abs(b),a=b or a=-b] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&\left| a\right| =\left| b\right| & \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=-b& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [x=abs(-2),x=2] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&x=\left| -2\right| & \cr \color{green}{\Leftrightarrow}&x=2& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [abs(a)=abs(b),a^2=b^2] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&\left| a\right| =\left| b\right| & \cr \color{green}{\Leftrightarrow}&a^2=b^2& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [x^2=9,x=#pm#3,x=3 or x=-3,x=3] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&x^2=9& \cr \color{green}{\Leftrightarrow}&x= \pm 3& \cr \color{green}{\Leftrightarrow}&x=3\,{\mbox{ or }}\, x=-3& \cr \color{green}{\Leftrightarrow}&x=3& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2=9,x=3] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&x^2=9& \cr \color{green}{\Leftrightarrow}&x=3& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [x^2=2,x=#pm#sqrt(2),x=sqrt(2) or x=-sqrt(2)] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&x^2=2& \cr \color{green}{\Leftrightarrow}&x= \pm \sqrt{2}& \cr \color{green}{\Leftrightarrow}&x=\sqrt{2}\,{\mbox{ or }}\, x=-\sqrt{2}& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2=2,x=sqrt(2)] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&x^2=2& \cr \color{green}{\Leftrightarrow}&x=\sqrt{2}& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [x^2 = a^2-b,x = sqrt(a^2-b)] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&x^2=a^2-b& \cr \color{green}{\Leftrightarrow}&x=\sqrt{a^2-b}& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR]
Equiv Pass [2*(x-3) = 4*x-3*(x+2),2*x-6=x-6,x=0] [] 1 1
\[\begin{array}{lll} &2\cdot \left(x-3\right)=4\cdot x-3\cdot \left(x+2\right)& \cr \color{green}{\Leftrightarrow}&2\cdot x-6=x-6& \cr \color{green}{\Leftrightarrow}&x=0& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [2*(x-3) = 5*x-3*(x+2),2*x-6=2*x-6,0=0,all] [] 1 1
\[\begin{array}{lll} &2\cdot \left(x-3\right)=5\cdot x-3\cdot \left(x+2\right)& \cr \color{green}{\Leftrightarrow}&2\cdot x-6=2\cdot x-6& \cr \color{green}{\Leftrightarrow}&0=0& \cr \color{green}{\Leftrightarrow}&\mathbb{R}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [2*(x-3) = 5*x-3*(x+1),2*x-6=2*x-3,0=3,{}] [] 1 1
\[\begin{array}{lll} &2\cdot \left(x-3\right)=5\cdot x-3\cdot \left(x+1\right)& \cr \color{green}{\Leftrightarrow}&2\cdot x-6=2\cdot x-3& \cr \color{green}{\Leftrightarrow}&0=3& \cr \color{green}{\Leftrightarrow}&\left \{ \right \}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a^2=b^2,a^2-b^2=0,(a-b)*(a+b)=0,a=b or a=-b] [] 1 1
\[\begin{array}{lll} &a^2=b^2& \cr \color{green}{\Leftrightarrow}&a^2-b^2=0& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a+b\right)=0& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=-b& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a^3=b^3,a^3-b^3=0,(a-b)*(a^2+a*b+b^2)=0,(a-b)=0,a=b] [] 0 0
\[\begin{array}{lll} &a^3=b^3& \cr \color{green}{\Leftrightarrow}&a^3-b^3=0& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a^2+a\cdot b+b^2\right)=0& \cr \color{red}{\Leftarrow}&a-b=0& \cr \color{green}{\Leftrightarrow}&a=b& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR]
Equiv Pass [a^3=b^3,a^3-b^3=0,(a-b)*(a^2+a*b+b^2)=0,(a-b)=0 or (a^2+a*b+b^2)=0, a=b or (a+(1+%i*sqrt(3))/2*b)*(a+(1-%i*sqrt(3))/2*b)=0, a=b or a=-(1+%i*sqrt(3))/2*b or a=-(1-%i*sqrt(3))/2*b] [] 1 1
\[\begin{array}{lll} &a^3=b^3& \cr \color{green}{\Leftrightarrow}&a^3-b^3=0& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a^2+a\cdot b+b^2\right)=0& \cr \color{green}{\Leftrightarrow}&a-b=0\,{\mbox{ or }}\, a^2+a\cdot b+b^2=0& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, \left(a+\frac{1+\mathrm{i}\cdot \sqrt{3}}{2}\cdot b\right)\cdot \left(a+\frac{1-\mathrm{i}\cdot \sqrt{3}}{2}\cdot b\right)=0& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=\frac{-\left(1+\mathrm{i}\cdot \sqrt{3}\right)}{2}\cdot b\,{\mbox{ or }}\, a=\frac{-\left(1-\mathrm{i}\cdot \sqrt{3}\right)}{2}\cdot b& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2-x=30,x^2-x-30=0,(x-6)*(x+5)=0,x-6=0 or x+5=0,x=6 or x=-5] [] 1 1
\[\begin{array}{lll} &x^2-x=30& \cr \color{green}{\Leftrightarrow}&x^2-x-30=0& \cr \color{green}{\Leftrightarrow}&\left(x-6\right)\cdot \left(x+5\right)=0& \cr \color{green}{\Leftrightarrow}&x-6=0\,{\mbox{ or }}\, x+5=0& \cr \color{green}{\Leftrightarrow}&x=6\,{\mbox{ or }}\, x=-5& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2=2,x^2-2=0,(x-sqrt(2))*(x+sqrt(2))=0,x=sqrt(2) or x=-sqrt(2)] [] 1 1
\[\begin{array}{lll} &x^2=2& \cr \color{green}{\Leftrightarrow}&x^2-2=0& \cr \color{green}{\Leftrightarrow}&\left(x-\sqrt{2}\right)\cdot \left(x+\sqrt{2}\right)=0& \cr \color{green}{\Leftrightarrow}&x=\sqrt{2}\,{\mbox{ or }}\, x=-\sqrt{2}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2=2,x=#pm#sqrt(2),x=sqrt(2) or x=-sqrt(2)] [] 1 1
\[\begin{array}{lll} &x^2=2& \cr \color{green}{\Leftrightarrow}&x= \pm \sqrt{2}& \cr \color{green}{\Leftrightarrow}&x=\sqrt{2}\,{\mbox{ or }}\, x=-\sqrt{2}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [(2*x-7)^2=(x+1)^2,(2*x-7)^2 -(x+1)^2=0,(2*x-7+x+1)*(2*x-7-x-1)=0,(3*x-6)*(x-8)=0,x=2 or x=8] [] 1 1
\[\begin{array}{lll} &{\left(2\cdot x-7\right)}^2={\left(x+1\right)}^2& \cr \color{green}{\Leftrightarrow}&{\left(2\cdot x-7\right)}^2-{\left(x+1\right)}^2=0& \cr \color{green}{\Leftrightarrow}&\left(2\cdot x-7+x+1\right)\cdot \left(2\cdot x-7-x-1\right)=0& \cr \color{green}{\Leftrightarrow}&\left(3\cdot x-6\right)\cdot \left(x-8\right)=0& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=8& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2-6*x=-9,(x-3)^2=0,x-3=0,x=3] [] 1 1
\[\begin{array}{lll} &x^2-6\cdot x=-9& \cr \color{green}{\Leftrightarrow}&{\left(x-3\right)}^2=0& \cr \color{green}{\mbox{(Same roots)}}&x-3=0& \cr \color{green}{\Leftrightarrow}&x=3& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR,SAMEROOTS, EQUIVCHAR]
Equiv Pass [(2*x-7)^2=(x+1)^2,sqrt((2*x-7)^2)=sqrt((x+1)^2),2*x-7=x+1,x=8] [] 0 0
\[\begin{array}{lll} &{\left(2\cdot x-7\right)}^2={\left(x+1\right)}^2& \cr \color{green}{\Leftrightarrow}&\sqrt{{\left(2\cdot x-7\right)}^2}=\sqrt{{\left(x+1\right)}^2}& \cr \color{red}{\Leftarrow}&2\cdot x-7=x+1& \cr \color{green}{\Leftrightarrow}&x=8& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR]
Equiv Pass [x^2-10*x+9 = 0, (x-5)^2-16 = 0, (x-5)^2 =16, x-5 =#pm#4, x-5 =4 or x-5=-4, x = 1 or x = 9] [] 1 1
\[\begin{array}{lll} &x^2-10\cdot x+9=0& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2-16=0& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2=16& \cr \color{green}{\Leftrightarrow}&x-5= \pm 4& \cr \color{green}{\Leftrightarrow}&x-5=4\,{\mbox{ or }}\, x-5=-4& \cr \color{green}{\Leftrightarrow}&x=1\,{\mbox{ or }}\, x=9& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2-2*p*x-q=0,x^2-2*p*x=q,x^2-2*p*x+p^2=q+p^2,(x-p)^2=q+p^2,x-p=#pm#sqrt(q+p^2),x-p=sqrt(q+p^2) or x-p=-sqrt(q+p^2),x=p+sqrt(q+p^2) or x=p-sqrt(q+p^2)] [] 1 1
\[\begin{array}{lll} &x^2-2\cdot p\cdot x-q=0& \cr \color{green}{\Leftrightarrow}&x^2-2\cdot p\cdot x=q& \cr \color{green}{\Leftrightarrow}&x^2-2\cdot p\cdot x+p^2=q+p^2& \cr \color{green}{\Leftrightarrow}&{\left(x-p\right)}^2=q+p^2& \cr \color{green}{\Leftrightarrow}&x-p= \pm \sqrt{q+p^2}& \cr \color{green}{\Leftrightarrow}&x-p=\sqrt{q+p^2}\,{\mbox{ or }}\, x-p=-\sqrt{q+p^2}& \cr \color{green}{\Leftrightarrow}&x=p+\sqrt{q+p^2}\,{\mbox{ or }}\, x=p-\sqrt{q+p^2}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2-10*x+7=0,(x-5)^2-18=0,(x-5)^2=sqrt(18)^2,(x-5)^2-sqrt(18)^2=0,(x-5-sqrt(18))*(x-5+sqrt(18))=0,x=5-sqrt(18) or x=5+sqrt(18)] [] 1 1
\[\begin{array}{lll} &x^2-10\cdot x+7=0& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2-18=0& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2={\it sqrt}^2\left(18\right)& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2-{\it sqrt}^2\left(18\right)=0& \cr \color{green}{\Leftrightarrow}&\left(x-5-\sqrt{18}\right)\cdot \left(x-5+\sqrt{18}\right)=0& \cr \color{green}{\Leftrightarrow}&x=5-\sqrt{18}\,{\mbox{ or }}\, x=5+\sqrt{18}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2+2*a*x = 0, x*(x+2*a)=0, (x+a-a)*(x+a+a)=0, (x+a)^2-a^2=0] [] 1 1
\[\begin{array}{lll} &x^2+2\cdot a\cdot x=0& \cr \color{green}{\Leftrightarrow}&x\cdot \left(x+2\cdot a\right)=0& \cr \color{green}{\Leftrightarrow}&\left(x+a-a\right)\cdot \left(x+a+a\right)=0& \cr \color{green}{\Leftrightarrow}&{\left(x+a\right)}^2-a^2=0& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^3-1=0,(x-1)*(x^2+x+1)=0,x=1] [] 0 0
\[\begin{array}{lll} &x^3-1=0& \cr \color{green}{\Leftrightarrow}&\left(x-1\right)\cdot \left(x^2+x+1\right)=0& \cr \color{red}{\Leftarrow}&x=1& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR,IMPLIEDCHAR]
Equiv Pass [x^3-1=0,(x-1)*(x^2+x+1)=0,x=1 or x^2+x+1=0,x=1 or x = -(sqrt(3)*%i+1)/2 or x=(sqrt(3)*%i-1)/2] [] 1 1
\[\begin{array}{lll} &x^3-1=0& \cr \color{green}{\Leftrightarrow}&\left(x-1\right)\cdot \left(x^2+x+1\right)=0& \cr \color{green}{\Leftrightarrow}&x=1\,{\mbox{ or }}\, x^2+x+1=0& \cr \color{green}{\Leftrightarrow}&x=1\,{\mbox{ or }}\, x=\frac{-\left(\sqrt{3}\cdot \mathrm{i}+1\right)}{2}\,{\mbox{ or }}\, x=\frac{\sqrt{3}\cdot \mathrm{i}-1}{2}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a*x^2+b*x+c=0 or a=0,a^2*x^2+a*b*x+a*c=0,(a*x)^2+b*(a*x)+a*c=0, (a*x)^2+b*(a*x)+b^2/4-b^2/4+a*c=0,(a*x+b/2)^2-b^2/4+a*c=0,(a*x+b/2)^2=b^2/4-a*c, a*x+b/2= #pm#sqrt(b^2/4-a*c),a*x=-b/2+sqrt(b^2/4-a*c) or a*x=-b/2-sqrt(b^2/4-a*c), (a=0 or x=(-b+sqrt(b^2-4*a*c))/(2*a)) or (a=0 or x=(-b-sqrt(b^2-4*a*c))/(2*a)), a^2=0 or x=(-b+sqrt(b^2-4*a*c))/(2*a) or x=(-b-sqrt(b^2-4*a*c))/(2*a)] [] 1 1
\[\begin{array}{lll} &a\cdot x^2+b\cdot x+c=0\,{\mbox{ or }}\, a=0& \cr \color{green}{\Leftrightarrow}&a^2\cdot x^2+a\cdot b\cdot x+a\cdot c=0& \cr \color{green}{\Leftrightarrow}&{\left(a\cdot x\right)}^2+b\cdot \left(a\cdot x\right)+a\cdot c=0& \cr \color{green}{\Leftrightarrow}&{\left(a\cdot x\right)}^2+b\cdot \left(a\cdot x\right)+\frac{b^2}{4}-\frac{b^2}{4}+a\cdot c=0& \cr \color{green}{\Leftrightarrow}&{\left(a\cdot x+\frac{b}{2}\right)}^2-\frac{b^2}{4}+a\cdot c=0& \cr \color{green}{\Leftrightarrow}&{\left(a\cdot x+\frac{b}{2}\right)}^2=\frac{b^2}{4}-a\cdot c& \cr \color{green}{\Leftrightarrow}&a\cdot x+\frac{b}{2}= \pm \sqrt{\frac{b^2}{4}-a\cdot c}& \cr \color{green}{\Leftrightarrow}&a\cdot x=-\frac{b}{2}+\sqrt{\frac{b^2}{4}-a\cdot c}\,{\mbox{ or }}\, a\cdot x=-\frac{b}{2}-\sqrt{\frac{b^2}{4}-a\cdot c}& \cr \color{green}{\Leftrightarrow}&a=0\,{\mbox{ or }}\, x=\frac{-b+\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}\,{\mbox{ or }}\, \left(a=0\,{\mbox{ or }}\, x=\frac{-b-\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}\right)& \cr \color{green}{\Leftrightarrow}&a^2=0\,{\mbox{ or }}\, x=\frac{-b+\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}\,{\mbox{ or }}\, x=\frac{-b-\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a*x^2+b*x=-c,4*a^2*x^2+4*a*b*x+b^2=b^2-4*a*c,(2*a*x+b)^2=b^2-4*a*c,2*a*x+b=#pm#sqrt(b^2-4*a*c),2*a*x=-b#pm#sqrt(b^2-4*a*c),x=(-b#pm#sqrt(b^2-4*a*c))/(2*a)] [] 0 0
\[\begin{array}{lll} &a\cdot x^2+b\cdot x=-c& \cr \color{red}{\Rightarrow}&4\cdot a^2\cdot x^2+4\cdot a\cdot b\cdot x+b^2=b^2-4\cdot a\cdot c& \cr \color{green}{\Leftrightarrow}&{\left(2\cdot a\cdot x+b\right)}^2=b^2-4\cdot a\cdot c& \cr \color{green}{\Leftrightarrow}&2\cdot a\cdot x+b= \pm \sqrt{b^2-4\cdot a\cdot c}& \cr \color{green}{\Leftrightarrow}&2\cdot a\cdot x={-b \pm \sqrt{b^2-4\cdot a\cdot c}}& \cr \color{red}{?}&x=\frac{{-b \pm \sqrt{b^2-4\cdot a\cdot c}}}{2\cdot a}& \cr \end{array}\]
[EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR]
Equiv Pass [a*x^2+b*x=-c or a=0,4*a^2*x^2+4*a*b*x+b^2=b^2-4*a*c,(2*a*x+b)^2=b^2-4*a*c,2*a*x+b=#pm#sqrt(b^2-4*a*c),2*a*x=-b#pm#sqrt(b^2-4*a*c),x=(-b#pm#sqrt(b^2-4*a*c))/(2*a) or a=0] [] 1 1
\[\begin{array}{lll} &a\cdot x^2+b\cdot x=-c\,{\mbox{ or }}\, a=0& \cr \color{green}{\Leftrightarrow}&4\cdot a^2\cdot x^2+4\cdot a\cdot b\cdot x+b^2=b^2-4\cdot a\cdot c& \cr \color{green}{\Leftrightarrow}&{\left(2\cdot a\cdot x+b\right)}^2=b^2-4\cdot a\cdot c& \cr \color{green}{\Leftrightarrow}&2\cdot a\cdot x+b= \pm \sqrt{b^2-4\cdot a\cdot c}& \cr \color{green}{\Leftrightarrow}&2\cdot a\cdot x={-b \pm \sqrt{b^2-4\cdot a\cdot c}}& \cr \color{green}{\Leftrightarrow}&x=\frac{{-b \pm \sqrt{b^2-4\cdot a\cdot c}}}{2\cdot a}\,{\mbox{ or }}\, a=0& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [sqrt(3*x+4) = 2+sqrt(x+2), 3*x+4=4+4*sqrt(x+2)+(x+2),x-1=2*sqrt(x+2),x^2-2*x+1 = 4*x+8,x^2-6*x-7 = 0,(x-7)*(x+1) = 0,x=7 or x=-1] [] 0 0
\[\begin{array}{lll} &\sqrt{3\cdot x+4}=2+\sqrt{x+2}&{\color{blue}{{x \in {\left[ -\frac{4}{3},\, \infty \right)}}}}\cr \color{red}{\Rightarrow}&3\cdot x+4=4+4\cdot \sqrt{x+2}+\left(x+2\right)&{\color{blue}{{x \in {\left[ -2,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&x-1=2\cdot \sqrt{x+2}&{\color{blue}{{x \in {\left[ -2,\, \infty \right)}}}}\cr \color{red}{\Rightarrow}&x^2-2\cdot x+1=4\cdot x+8& \cr \color{green}{\Leftrightarrow}&x^2-6\cdot x-7=0& \cr \color{green}{\Leftrightarrow}&\left(x-7\right)\cdot \left(x+1\right)=0& \cr \color{green}{\Leftrightarrow}&x=7\,{\mbox{ or }}\, x=-1& \cr \end{array}\]
[EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [sqrt(3*x+4) = 2+sqrt(x+2), 3*x+4=4+4*sqrt(x+2)+(x+2),x-1=2*sqrt(x+2),x^2-2*x+1 = 4*x+8,x^2-6*x-7 = 0,(x-7)*(x+1) = 0,x=7 or x=-1,x=7] [] [assumepos] 1 1
\[\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&\sqrt{3\cdot x+4}=2+\sqrt{x+2}&{\color{blue}{{x \in {\left[ 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&3\cdot x+4=4+4\cdot \sqrt{x+2}+\left(x+2\right)&{\color{blue}{{x \in {\left[ 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&x-1=2\cdot \sqrt{x+2}&{\color{blue}{{x \in {\left[ 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&x^2-2\cdot x+1=4\cdot x+8& \cr \color{green}{\Leftrightarrow}&x^2-6\cdot x-7=0& \cr \color{green}{\Leftrightarrow}&\left(x-7\right)\cdot \left(x+1\right)=0& \cr \color{green}{\Leftrightarrow}&x=7\,{\mbox{ or }}\, x=-1& \cr \color{green}{\Leftrightarrow}&x=7& \cr \end{array}\]
[ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x*(x-1)*(x-2)=0,x*(x-1)=0,x*(x-1)*(x-2)=0,x*(x^2-2)=0] [] 0 0
\[\begin{array}{lll} &x\cdot \left(x-1\right)\cdot \left(x-2\right)=0& \cr \color{red}{\Leftarrow}&x\cdot \left(x-1\right)=0& \cr \color{red}{\Rightarrow}&x\cdot \left(x-1\right)\cdot \left(x-2\right)=0& \cr \color{red}{?}&x\cdot \left(x^2-2\right)=0& \cr \end{array}\]
[EMPTYCHAR,IMPLIEDCHAR,IMPLIESCHAR,QMCHAR]
Equiv Pass [x^2-6*x=-9,x=3] [] 1 1
\[\begin{array}{lll} &x^2-6\cdot x=-9& \cr \color{green}{\mbox{(Same roots)}}&x=3& \cr \end{array}\]
[EMPTYCHAR,SAMEROOTS]
Equiv Pass [x=1 nounor x=-2 nounor x=1,x^3-3*x=-2,x=1 nounor x=-2] [] 1 1
\[\begin{array}{lll} &x=1\,{\mbox{ or }}\, x=-2\,{\mbox{ or }}\, x=1& \cr \color{green}{\Leftrightarrow}&x^3-3\cdot x=-2& \cr \color{green}{\mbox{(Same roots)}}&x=1\,{\mbox{ or }}\, x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR,SAMEROOTS]
Equiv Pass [9*x^3-24*x^2+13*x=2,x=1/3 nounor x=2] [] 1 1
\[\begin{array}{lll} &9\cdot x^3-24\cdot x^2+13\cdot x=2& \cr \color{green}{\mbox{(Same roots)}}&x=\frac{1}{3}\,{\mbox{ or }}\, x=2& \cr \end{array}\]
[EMPTYCHAR,SAMEROOTS]
Equiv Pass [(x-2)^43*(x+1/3)^60=0,(3*x+1)^4*(x-2)^2=0,x=-1/3 nounor x=2] [] 1 1
\[\begin{array}{lll} &{\left(x-2\right)}^{43}\cdot {\left(x+\frac{1}{3}\right)}^{60}=0& \cr \color{green}{\mbox{(Same roots)}}&{\left(3\cdot x+1\right)}^4\cdot {\left(x-2\right)}^2=0& \cr \color{green}{\mbox{(Same roots)}}&x=\frac{-1}{3}\,{\mbox{ or }}\, x=2& \cr \end{array}\]
[EMPTYCHAR,SAMEROOTS,SAMEROOTS]
Equiv Pass [2^x=4,x*log(2)=log(4),x=log(2^2)/log(2),x=2*log(2)/log(2),x=2] [] 1 1
\[\begin{array}{lll} &2^{x}=4& \cr \color{green}{\Leftrightarrow}&x\cdot \ln \left( 2 \right)=\ln \left( 4 \right)& \cr \color{green}{\Leftrightarrow}&x=\frac{\ln \left( 2^2 \right)}{\ln \left( 2 \right)}& \cr \color{green}{\Leftrightarrow}&x=\frac{2\cdot \ln \left( 2 \right)}{\ln \left( 2 \right)}& \cr \color{green}{\Leftrightarrow}&x=2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^log(y),stackeq(e^(log(x)*log(y))),stackeq(e^(log(y)*log(x))),stackeq(y^log(x))] [] 1 1
\[\begin{array}{lll} &x^{\ln \left( y \right)}& \cr \color{green}{\checkmark}&=e^{\ln \left( x \right)\cdot \ln \left( y \right)}& \cr \color{green}{\checkmark}&=e^{\ln \left( y \right)\cdot \ln \left( x \right)}& \cr \color{green}{\checkmark}&=y^{\ln \left( x \right)}& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK]
Equiv Pass [lg(x+17,3)-2=lg(2*x,3),lg(x+17,3)-lg(2*x,3)=2,lg((x+17)/(2*x),3)=2,(x+17)/(2*x)=3^2,(x+17)=18*x,17*x=17,x=1] [] 1 1
\[\begin{array}{lll} &\log_{3}\left(x+17\right)-2=\log_{3}\left(2\cdot x\right)&{\color{blue}{{x \in {\left( 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&\log_{3}\left(x+17\right)-\log_{3}\left(2\cdot x\right)=2&{\color{blue}{{x \in {\left( 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&\log_{3}\left(\frac{x+17}{2\cdot x}\right)=2& \cr \color{green}{\log(?)}&\frac{x+17}{2\cdot x}=3^2&{\color{blue}{{x \not\in {\left \{0 \right \}}}}}\cr \color{green}{\Leftrightarrow}&x+17=18\cdot x& \cr \color{green}{\Leftrightarrow}&17\cdot x=17& \cr \color{green}{\Leftrightarrow}&x=1& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,EQUIVLOG, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x=(1+y/n)^n,x^(1/n)=(1+y/n),y/n=x^(1/n)-1,y=n*(x^(1/n)-1)] [] 0 0
\[\begin{array}{lll} &x={\left(1+\frac{y}{n}\right)}^{n}& \cr \color{red}{?}&x^{\frac{1}{n}}=1+\frac{y}{n}& \cr \color{green}{\Leftrightarrow}&\frac{y}{n}=x^{\frac{1}{n}}-1& \cr \color{green}{\Leftrightarrow}&y=n\cdot \left(x^{\frac{1}{n}}-1\right)& \cr \end{array}\]
[EMPTYCHAR,QMCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a^3=b^3,a^3-b^3=0,(a-b)*(a^2+a*b+b^2)=0,(a-b)=0,a=b] [] [assumereal] 0 0
\[\begin{array}{lll}\color{blue}{(\mathbb{R})}&a^3=b^3& \cr \color{green}{\Leftrightarrow}&a^3-b^3=0& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a^2+a\cdot b+b^2\right)=0& \cr \color{red}{\Leftarrow}&a-b=0& \cr \color{green}{\Leftrightarrow}&a=b& \cr \end{array}\]
[ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR]
Equiv Pass [x^3-1=0,(x-1)*(x^2+x+1)=0,x=1] [] [assumereal] 1 1
\[\begin{array}{lll}\color{blue}{(\mathbb{R})}&x^3-1=0& \cr \color{green}{\Leftrightarrow}&\left(x-1\right)\cdot \left(x^2+x+1\right)=0& \cr \color{green}{\Leftrightarrow}\, \color{blue}{(\mathbb{R})}&x=1& \cr \end{array}\]
[ASSUMEREALVARS, EQUIVCHAR, EQUIVCHARREAL]
Equiv Pass [x^4=2,x^4-2=0,(x^2-sqrt(2))*(x^2+sqrt(2))=0,x^2=sqrt(2),x=#pm# 2^(1/4)] [] [assumereal] 1 1
\[\begin{array}{lll}\color{blue}{(\mathbb{R})}&x^4=2& \cr \color{green}{\Leftrightarrow}&x^4-2=0& \cr \color{green}{\Leftrightarrow}&\left(x^2-\sqrt{2}\right)\cdot \left(x^2+\sqrt{2}\right)=0& \cr \color{green}{\Leftrightarrow}\, \color{blue}{(\mathbb{R})}&x^2=\sqrt{2}& \cr \color{green}{\Leftrightarrow}&x= \pm 2^{\frac{1}{4}}& \cr \end{array}\]
[ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHARREAL, EQUIVCHAR]
Equiv Pass [6*x-12=3*(x-2),6*x-12+3*(x-2)=0,9*x-18=0,x=2] [] 1 1
\[\begin{array}{lll} &6\cdot x-12=3\cdot \left(x-2\right)& \cr \color{green}{\Leftrightarrow}&6\cdot x-12+3\cdot \left(x-2\right)=0& \cr \color{green}{\Leftrightarrow}&9\cdot x-18=0& \cr \color{green}{\Leftrightarrow}&x=2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2-6*x+9=0,x^2-6*x=-9,x*(x-6)=3*-3,x=3 or x-6=-3,x=3] [] 1 1
\[\begin{array}{lll} &x^2-6\cdot x+9=0& \cr \color{green}{\Leftrightarrow}&x^2-6\cdot x=-9& \cr \color{green}{\Leftrightarrow}&x\cdot \left(x-6\right)=3\cdot \left(-3\right)& \cr \color{green}{\Leftrightarrow}&x=3\,{\mbox{ or }}\, x-6=-3& \cr \color{green}{\mbox{(Same roots)}}&x=3& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,SAMEROOTS]
Equiv Pass [(x+3)*(2-x)=4,x+3=4 or (2-x)=4,x=1 or x=-2] [] 1 1
\[\begin{array}{lll} &\left(x+3\right)\cdot \left(2-x\right)=4& \cr \color{green}{\Leftrightarrow}&x+3=4\,{\mbox{ or }}\, 2-x=4& \cr \color{green}{\Leftrightarrow}&x=1\,{\mbox{ or }}\, x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [(x-p)*(x-q)=0,x^2-p*x-q*x+p*q=0,1+q-x-p-p*q+p*x+x+q*x-x^2=1-p+q,(1+q-x)*(1-p+x)=1-p+q,(1+q-x)=1-p+q or (1-p+x)=1-p+q,x=p or x=q] [] 1 1
\[\begin{array}{lll} &\left(x-p\right)\cdot \left(x-q\right)=0& \cr \color{green}{\Leftrightarrow}&x^2-p\cdot x+\left(-q\right)\cdot x+p\cdot q=0& \cr \color{green}{\Leftrightarrow}&1+q-x-p+\left(-p\right)\cdot q+p\cdot x+x+q\cdot x-x^2=1-p+q& \cr \color{green}{\Leftrightarrow}&\left(1+q-x\right)\cdot \left(1-p+x\right)=1-p+q& \cr \color{green}{\Leftrightarrow}&1+q-x=1-p+q\,{\mbox{ or }}\, 1-p+x=1-p+q& \cr \color{green}{\Leftrightarrow}&x=p\,{\mbox{ or }}\, x=q& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a=b, a^2=a*b, a^2-b^2=a*b-b^2, (a-b)*(a+b)=b*(a-b), a+b=b, 2*a=a, 1=2] [] 0 0
\[\begin{array}{lll} &a=b& \cr \color{red}{\Rightarrow}&a^2=a\cdot b& \cr \color{green}{\Leftrightarrow}&a^2-b^2=a\cdot b-b^2& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a+b\right)=b\cdot \left(a-b\right)& \cr \color{red}{\Leftarrow}&a+b=b& \cr \color{green}{\Leftrightarrow}&2\cdot a=a& \cr \color{red}{\Leftarrow}&1=2& \cr \end{array}\]
[EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR,IMPLIEDCHAR]
Equiv Pass [a=b or a=0, a^2=a*b, a^2-b^2=a*b-b^2, (a-b)*(a+b)=b*(a-b), a+b=b or a-b=0, 2*a=a or a=b, 2=1 or a=0 or a=b, a=0 or a=b] [] 1 1
\[\begin{array}{lll} &a=b\,{\mbox{ or }}\, a=0& \cr \color{green}{\Leftrightarrow}&a^2=a\cdot b& \cr \color{green}{\Leftrightarrow}&a^2-b^2=a\cdot b-b^2& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a+b\right)=b\cdot \left(a-b\right)& \cr \color{green}{\Leftrightarrow}&a+b=b\,{\mbox{ or }}\, a-b=0& \cr \color{green}{\Leftrightarrow}&2\cdot a=a\,{\mbox{ or }}\, a=b& \cr \color{green}{\Leftrightarrow}&2=1\,{\mbox{ or }}\, a=0\,{\mbox{ or }}\, a=b& \cr \color{green}{\Leftrightarrow}&a=0\,{\mbox{ or }}\, a=b& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [(x^2-4)/(x-2)=0,(x-2)*(x+2)/(x-2)=0,x+2=0,x=-2] [] 1 1
\[\begin{array}{lll} &\frac{x^2-4}{x-2}=0&{\color{blue}{{x \not\in {\left \{2 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{\left(x-2\right)\cdot \left(x+2\right)}{x-2}=0& \cr \color{green}{\Leftrightarrow}&x+2=0& \cr \color{green}{\Leftrightarrow}&x=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [(x^2-4)/(x-2)=0,(x^2-4)=0,(x-2)*(x+2)=0,x=-2 or x=2] [] 0 0
\[\begin{array}{lll} &\frac{x^2-4}{x-2}=0&{\color{blue}{{x \not\in {\left \{2 \right \}}}}}\cr \color{red}{\Rightarrow}&x^2-4=0& \cr \color{green}{\Leftrightarrow}&\left(x-2\right)\cdot \left(x+2\right)=0& \cr \color{green}{\Leftrightarrow}&x=-2\,{\mbox{ or }}\, x=2& \cr \end{array}\]
[EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [5*x/(2*x+1)-3/(x+1) = 1,5*x*(x+1)-3*(2*x+1)=(x+1)*(2*x+1),5*x^2+5*x-6*x-3=2*x^2+3*x+1,3*x^2-4*x-4=0,(x-2)*(3*x+2)=0,x=2 or x=-2/3] [] 1 1
\[\begin{array}{lll} &\frac{5\cdot x}{2\cdot x+1}-\frac{3}{x+1}=1&{\color{blue}{{x \not\in {\left \{-1 , -\frac{1}{2} \right \}}}}}\cr \color{green}{\Leftrightarrow}&5\cdot x\cdot \left(x+1\right)-3\cdot \left(2\cdot x+1\right)=\left(x+1\right)\cdot \left(2\cdot x+1\right)& \cr \color{green}{\Leftrightarrow}&5\cdot x^2+5\cdot x-6\cdot x-3=2\cdot x^2+3\cdot x+1& \cr \color{green}{\Leftrightarrow}&3\cdot x^2-4\cdot x-4=0& \cr \color{green}{\Leftrightarrow}&\left(x-2\right)\cdot \left(3\cdot x+2\right)=0& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=\frac{-2}{3}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [(x+10)/(x-6)-5= (4*x-40)/(13-x),(x+10-5*(x-6))/(x-6)= (4*x-40)/(13-x), (4*x-40)/(6-x)= (4*x-40)/(13-x),6-x= 13-x,6= 13] [] 0 0
\[\begin{array}{lll} &\frac{x+10}{x-6}-5=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{6 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{x+10-5\cdot \left(x-6\right)}{x-6}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{6 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{4\cdot x-40}{6-x}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{6 , 13 \right \}}}}}\cr \color{red}{?}&6-x=13-x& \cr \color{green}{\Leftrightarrow}&6=13& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR, EQUIVCHAR]
Equiv Pass [(x+5)/(x-7)-5= (4*x-40)/(13-x),(x+5-5*(x-7))/(x-7)= (4*x-40)/(13-x), (4*x-40)/(7-x)= (4*x-40)/(13-x),7-x= 13-x,7= 13] [] 0 0
\[\begin{array}{lll} &\frac{x+5}{x-7}-5=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{x+5-5\cdot \left(x-7\right)}{x-7}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{4\cdot x-40}{7-x}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{red}{\Leftarrow}&7-x=13-x& \cr \color{green}{\Leftrightarrow}&7=13& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR]
Equiv Pass [(x+5)/(x-7)-5= (4*x-40)/(13-x),(x+5-5*(x-7))/(x-7)= (4*x-40)/(13-x), (4*x-40)/(7-x)= (4*x-40)/(13-x),7-x= 13-x or 4*x-40=0,7= 13 or 4*x=40,x=10] [] 1 1
\[\begin{array}{lll} &\frac{x+5}{x-7}-5=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{x+5-5\cdot \left(x-7\right)}{x-7}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{4\cdot x-40}{7-x}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&7-x=13-x\,{\mbox{ or }}\, 4\cdot x-40=0& \cr \color{green}{\Leftrightarrow}&7=13\,{\mbox{ or }}\, 4\cdot x=40& \cr \color{green}{\Leftrightarrow}&x=10& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a*x^2+b*x+c=0,a=0 nounand b=0 nounand c=0,a*x^2+b*x+c=0] [] 1 1
\[\begin{array}{lll} &a\cdot x^2+b\cdot x+c=0& \cr \color{green}{\equiv (\cdots ? x)}&\left\{\begin{array}{l}a=0\cr b=0\cr c=0\cr \end{array}\right.& \cr \color{green}{(\cdots ? x)\equiv}&a\cdot x^2+b\cdot x+c=0& \cr \end{array}\]
[EMPTYCHAR,EQUATECOEFFLOSS(x),EQUATECOEFFGAIN(x)]
Equiv Pass [a*x^2+b*x+c=A*x^2+B*x+C,a=A nounand b=B nounand c=C,a*x^2+b*x+c=A*x^2+B*x+C] [] 1 1
\[\begin{array}{lll} &a\cdot x^2+b\cdot x+c=A\cdot x^2+B\cdot x+C& \cr \color{green}{\equiv (\cdots ? x)}&\left\{\begin{array}{l}a=A\cr b=B\cr c=C\cr \end{array}\right.& \cr \color{green}{(\cdots ? x)\equiv}&a\cdot x^2+b\cdot x+c=A\cdot x^2+B\cdot x+C& \cr \end{array}\]
[EMPTYCHAR,EQUATECOEFFLOSS(x),EQUATECOEFFGAIN(x)]
Equiv Pass [(x-1)*(x+4), stackeq(x^2-x+4*x-4),stackeq(x^2+3*x-4)] [] 1 1
\[\begin{array}{lll} &\left(x-1\right)\cdot \left(x+4\right)& \cr \color{green}{\checkmark}&=x^2-x+4\cdot x-4& \cr \color{green}{\checkmark}&=x^2+3\cdot x-4& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK]
Equiv Pass [(x-1)*(x+4), stackeq(x^2-x+4*x-4),stackeq(x^2+3*x-4)] [] 1 1
\[\begin{array}{lll} &\left(x-1\right)\cdot \left(x+4\right)& \cr \color{green}{\checkmark}&=x^2-x+4\cdot x-4& \cr \color{green}{\checkmark}&=x^2+3\cdot x-4& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK]
Equiv Pass [x^2-2,stackeq((x-sqrt(2))*(x+sqrt(2)))] [] 1 1
\[\begin{array}{lll} &x^2-2& \cr \color{green}{\checkmark}&=\left(x-\sqrt{2}\right)\cdot \left(x+\sqrt{2}\right)& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK]
Equiv Pass [x^2+4,stackeq((x-2*i)*(x+2*i))] [] 1 1
\[\begin{array}{lll} &x^2+4& \cr \color{green}{\checkmark}&=\left(x-2\cdot \mathrm{i}\right)\cdot \left(x+2\cdot \mathrm{i}\right)& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK]
Equiv Pass [x^2+2*a*x,x^2+2*a*x+a^2-a^2,(x+a)^2-a^2] [] 1 1
\[\begin{array}{lll} &x^2+2\cdot a\cdot x& \cr \color{green}{\Leftrightarrow}&x^2+2\cdot a\cdot x+a^2-a^2& \cr \color{green}{\Leftrightarrow}&{\left(x+a\right)}^2-a^2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2+2*a*x,stackeq(x^2+2*a*x+a^2-a^2),stackeq((x+a)^2-a^2)] [] 1 1
\[\begin{array}{lll} &x^2+2\cdot a\cdot x& \cr \color{green}{\checkmark}&=x^2+2\cdot a\cdot x+a^2-a^2& \cr \color{green}{\checkmark}&={\left(x+a\right)}^2-a^2& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK]
Equiv Pass [(y-z)/(y*z)+(z-x)/(z*x)+(x-y)/(x*y),(x*(y-z)+y*(z-x)+z*(x-y))/(x*y*z),0] [] 1 1
\[\begin{array}{lll} &\frac{y-z}{y\cdot z}+\frac{z-x}{z\cdot x}+\frac{x-y}{x\cdot y}& \cr \color{green}{\Leftrightarrow}&\frac{x\cdot \left(y-z\right)+y\cdot \left(z-x\right)+z\cdot \left(x-y\right)}{x\cdot y\cdot z}& \cr \color{green}{\Leftrightarrow}&0& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [(y-z)/(y*z)+(z-x)/(z*x)+(x-y)/(x*y),stackeq((x*(y-z)+y*(z-x)+z*(x-y))/(x*y*z)),stackeq(0)] [] 1 1
\[\begin{array}{lll} &\frac{y-z}{y\cdot z}+\frac{z-x}{z\cdot x}+\frac{x-y}{x\cdot y}& \cr \color{green}{\checkmark}&=\frac{x\cdot \left(y-z\right)+y\cdot \left(z-x\right)+z\cdot \left(x-y\right)}{x\cdot y\cdot z}& \cr \color{green}{\checkmark}&=0& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK]
Equiv Pass [2*(a^2*b^2+b^2*c^2+c^2*a^2)-(a^4+b^4+c^4),stackeq(4*a^2*b^2-(a^4+b^4+c^4+2*a^2*b^2-2*b^2*c^2-2*c^2*a^2)),stackeq((2*a*b)^2-(b^2+a^2-c^2)^2,(2*a*b+b^2+a^2-c^2)*(2*a*b-b^2-a^2+c^2)),stackeq(((a+b)^2-c^2)*(c^2-(a-b)^2)),stackeq((a+b+c)*(a+b-c)*(c+a-b)*(c-a+b))] [] 1 1
\[\begin{array}{lll} &2\cdot \left(a^2\cdot b^2+b^2\cdot c^2+c^2\cdot a^2\right)-\left(a^4+b^4+c^4\right)& \cr \color{green}{\checkmark}&=4\cdot a^2\cdot b^2-\left(a^4+b^4+c^4+2\cdot a^2\cdot b^2-2\cdot b^2\cdot c^2-2\cdot c^2\cdot a^2\right)& \cr \color{green}{\checkmark}&={\left(2\cdot a\cdot b\right)}^2-{\left(b^2+a^2-c^2\right)}^2& \cr \color{green}{\checkmark}&=\left({\left(a+b\right)}^2-c^2\right)\cdot \left(c^2-{\left(a-b\right)}^2\right)& \cr \color{green}{\checkmark}&=\left(a+b+c\right)\cdot \left(a+b-c\right)\cdot \left(c+a-b\right)\cdot \left(c-a+b\right)& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK]
Equiv Pass [abs(x-1/2)+abs(x+1/2)-2,stackeq(abs(x)-1)] [] 0 0
\[\begin{array}{lll} &\left| x-\frac{1}{2}\right| +\left| x+\frac{1}{2}\right| -2& \cr \color{red}{?}&=\left| x\right| -1& \cr \end{array}\]
[EMPTYCHAR,QMCHAR]
Equiv Pass [11*sqrt(abs(x)+1)=25-x,11^2*(abs(x)+1)=(25-x)^2,11^2*abs(x)=(25-x)^2-11^2,11^4*x^2=((25-x)^2-11^2)^2, ((25-x)^2-11^2)^2-11^4*x^2=0,((25-x)^2-11^2-11^2*x)*((25-x)^2-11^2+11^2*x)=0,(x^2-50*x+504-121*x)*(x^2-50*x+504+121*x)=0, (x-168)*(x-3)*(x+8)*(x+63)=0] [] 0 0
\[\begin{array}{lll} &11\cdot \sqrt{\left| x\right| +1}=25-x& \cr \color{red}{?}&11^2\cdot \left(\left| x\right| +1\right)={\left(25-x\right)}^2& \cr \color{green}{\Leftrightarrow}&11^2\cdot \left| x\right| ={\left(25-x\right)}^2-11^2& \cr \color{green}{\Leftrightarrow}&11^4\cdot x^2={\left({\left(25-x\right)}^2-11^2\right)}^2& \cr \color{green}{\Leftrightarrow}&{\left({\left(25-x\right)}^2-11^2\right)}^2-11^4\cdot x^2=0& \cr \color{green}{\Leftrightarrow}&\left({\left(25-x\right)}^2-11^2+\left(-11^2\right)\cdot x\right)\cdot \left({\left(25-x\right)}^2-11^2+11^2\cdot x\right)=0& \cr \color{green}{\Leftrightarrow}&\left(x^2-50\cdot x+504-121\cdot x\right)\cdot \left(x^2-50\cdot x+504+121\cdot x\right)=0& \cr \color{green}{\Leftrightarrow}&\left(x-168\right)\cdot \left(x-3\right)\cdot \left(x+8\right)\cdot \left(x+63\right)=0& \cr \end{array}\]
[EMPTYCHAR,QMCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [1/(x^2+1)=1/((x+%i)*(x-%i)), stackeq(1/(2*%i)*(1/(x-%i)-1/(x+%i)))] [] 1 1
\[\begin{array}{lll}\color{green}{\checkmark}&\frac{1}{x^2+1}=\frac{1}{\left(x+\mathrm{i}\right)\cdot \left(x-\mathrm{i}\right)}& \cr \color{green}{\checkmark}&=\frac{1}{2\cdot \mathrm{i}}\cdot \left(\frac{1}{x-\mathrm{i}}-\frac{1}{x+\mathrm{i}}\right)& \cr \end{array}\]
[CHECKMARK, CHECKMARK]
Equiv Pass [((a-b)/(a^2+a*b))/((a^2-2*a*b+b^2)/(a^4-b^4)),stackeq(((a-b)*(a-b)*(a+b)*(a^2+b^2))/(a*(a+b)*(a-b)^2)),stackeq((a^2+b^2)/a),stackeq(a+b^2/a)] [] 1 1
\[\begin{array}{lll} &\frac{\frac{a-b}{a^2+a\cdot b}}{\frac{a^2-2\cdot a\cdot b+b^2}{a^4-b^4}}& \cr \color{green}{\checkmark}&=\frac{\left(a-b\right)\cdot \left(a-b\right)\cdot \left(a+b\right)\cdot \left(a^2+b^2\right)}{a\cdot \left(a+b\right)\cdot {\left(a-b\right)}^2}& \cr \color{green}{\checkmark}&=\frac{a^2+b^2}{a}& \cr \color{green}{\checkmark}&=a+\frac{b^2}{a}& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK]
Equiv Pass [sum(k,k,1,n+1),stackeq(sum(k,k,1,n)+(n+1)),stackeq(n*(n+1)/2 +n+1),stackeq((n+1)*(n+1+1)/2),stackeq((n+1)*(n+2)/2)] [] 1 1
\[\begin{array}{lll} &\sum_{k=1}^{n+1}{k}& \cr \color{green}{\checkmark}&=\sum_{k=1}^{n}{k}+\left(n+1\right)& \cr \color{green}{\checkmark}&=\frac{n\cdot \left(n+1\right)}{2}+n+1& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)\cdot \left(n+1+1\right)}{2}& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)\cdot \left(n+2\right)}{2}& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK]
Equiv Pass [log((a-1)^n*product(x_i^(-a),i,1,n)),stackeq(n*log(a-1)-a*sum(log(x_i),i,1,n))] [] 1 1
\[\begin{array}{lll} &\ln \left( {\left(a-1\right)}^{n}\cdot \prod_{i=1}^{n}{\frac{1}{{{x}_{i}}^{a}}} \right)& \cr \color{green}{\checkmark}&=n\cdot \ln \left( a-1 \right)-a\cdot \sum_{i=1}^{n}{\ln \left( {x}_{i} \right)}& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK]
Equiv Pass [binomial(n,k)+binomial(n,k+1),stackeq(n!/(k!*(n-k)!)+n!/((k+1)!*(n-k-1)!)),stackeq(n!/(k!*(n-k)*(n-k-1)!)+n!/((k+1)!*(n-k-1)!)),stackeq(n!/(k!*(n-k-1)!)*(1/(n-k)+1/(k+1))),stackeq(n!/(k!*(n-k-1)!)*((n+1)/((n-k)*(k+1)))),stackeq((n+1)*n!/(k!*(n-k-1)!)*(1/((k+1)*(n-k)))),stackeq((n+1)*n!/((k+1)*k!*(n-k)*(n-k-1)!)),stackeq(((n+1)!/((k+1)!)*(1/((n-k)*(n-k-1)!)))),stackeq((n+1)!/((k+1)!*(n-k)!)),stackeq(binomial(n+1,k+1))] [] 1 1
\[\begin{array}{lll} &{{n}\choose{k}}+{{n}\choose{k+1}}& \cr \color{green}{\checkmark}&=\frac{n!}{k!\cdot \left(n-k\right)!}+\frac{n!}{\left(k+1\right)!\cdot \left(n-k-1\right)!}& \cr \color{green}{\checkmark}&=\frac{n!}{k!\cdot \left(n-k\right)\cdot \left(n-k-1\right)!}+\frac{n!}{\left(k+1\right)!\cdot \left(n-k-1\right)!}& \cr \color{green}{\checkmark}&=\frac{n!}{k!\cdot \left(n-k-1\right)!}\cdot \left(\frac{1}{n-k}+\frac{1}{k+1}\right)& \cr \color{green}{\checkmark}&=\frac{n!}{k!\cdot \left(n-k-1\right)!}\cdot \left(\frac{n+1}{\left(n-k\right)\cdot \left(k+1\right)}\right)& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)\cdot n!}{k!\cdot \left(n-k-1\right)!}\cdot \left(\frac{1}{\left(k+1\right)\cdot \left(n-k\right)}\right)& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)\cdot n!}{\left(k+1\right)\cdot k!\cdot \left(n-k\right)\cdot \left(n-k-1\right)!}& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)!}{\left(k+1\right)!}\cdot \left(\frac{1}{\left(n-k\right)\cdot \left(n-k-1\right)!}\right)& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)!}{\left(k+1\right)!\cdot \left(n-k\right)!}& \cr \color{green}{\checkmark}&={{n+1}\choose{k+1}}& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK]
Equiv Pass [(x-1)^2=(x-1)*(x-1), stackeq(x^2-2*x+1)] [] 1 1
\[\begin{array}{lll}\color{green}{\checkmark}&{\left(x-1\right)}^2=\left(x-1\right)\cdot \left(x-1\right)& \cr \color{green}{\checkmark}&=x^2-2\cdot x+1& \cr \end{array}\]
[CHECKMARK, CHECKMARK]
Equiv Pass [(x-1)^2=(x-1)*(x-1), stackeq(x^2-2*x+2)] [] 0 0
\[\begin{array}{lll}\color{green}{\checkmark}&{\left(x-1\right)}^2=\left(x-1\right)\cdot \left(x-1\right)& \cr \color{red}{?}&=x^2-2\cdot x+2& \cr \end{array}\]
[CHECKMARK,QMCHAR]
Equiv Pass [(x-2)^2=(x-1)*(x-1), stackeq(x^2-2*x+1)] [] 0 0
\[\begin{array}{lll}\color{red}{?}&{\left(x-2\right)}^2=\left(x-1\right)\cdot \left(x-1\right)& \cr \color{green}{\checkmark}&=x^2-2\cdot x+1& \cr \end{array}\]
[QMCHAR, CHECKMARK]
Equiv Pass [4^((n+1)+1)-1= 4*4^(n+1)-1,stackeq(4*(4^(n+1)-1)+3)] [] 1 1
\[\begin{array}{lll}\color{green}{\checkmark}&4^{n+1+1}-1=4\cdot 4^{n+1}-1& \cr \color{green}{\checkmark}&=4\cdot \left(4^{n+1}-1\right)+3& \cr \end{array}\]
[CHECKMARK, CHECKMARK]
Equiv Pass [2*x+3*y=6 and 4*x+9*y=15,2*x+3*y=6 and -2*x=-3,3+3*y=6 and 2*x=3,y=1 and x=3/2] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}2\cdot x+3\cdot y=6\cr 4\cdot x+9\cdot y=15\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}2\cdot x+3\cdot y=6\cr -2\cdot x=-3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}3+3\cdot y=6\cr 2\cdot x=3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}y=1\cr x=\frac{3}{2}\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [2*x+3*y=6 and 4*x+9*y=15,2*x+3*y=6 and -2*x=-3,3+3*y=6 and 2*x=3,y=1 and x=3] [] 0 0
\[\begin{array}{lll} &\left\{\begin{array}{l}2\cdot x+3\cdot y=6\cr 4\cdot x+9\cdot y=15\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}2\cdot x+3\cdot y=6\cr -2\cdot x=-3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}3+3\cdot y=6\cr 2\cdot x=3\cr \end{array}\right.& \cr \color{red}{?}&\left\{\begin{array}{l}y=1\cr x=3\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR]
Equiv Pass [x^2+y^2=8 and x=y, 2*x^2=8 and y=x, x^2=4 and y=x, x= #pm#2 and y=x, (x= 2 and y=x) or (x=-2 and y=x), (x=2 and y=2) or (x=-2 and y=-2)] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}x^2+y^2=8\cr x=y\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}2\cdot x^2=8\cr y=x\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x^2=4\cr y=x\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x= \pm 2\cr y=x\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ and }}\, y=x\,{\mbox{ or }}\, x=-2\,{\mbox{ and }}\, y=x& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ and }}\, y=2\,{\mbox{ or }}\, x=-2\,{\mbox{ and }}\, y=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2+y^2=5 and x*y=2, x^2+y^2-5=0 and x*y-2=0, x^2-2*x*y+y^2-1=0 and x^2+2*x*y+y^2-9=0, (x-y)^2-1=0 and (x+y)^2-3^2=0, (x-y=1 and x+y=3) or (x-y=-1 and x+y=3) or (x-y=1 and x+y=-3) or (x-y=-1 and x+y=-3), (x=1 and y=2) or (x=2 and y=1) or (x=-2 and y=-1) or (x=-1 and y=-2)] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}x^2+y^2=5\cr x\cdot y=2\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x^2+y^2-5=0\cr x\cdot y-2=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x^2-2\cdot x\cdot y+y^2-1=0\cr x^2+2\cdot x\cdot y+y^2-9=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}{\left(x-y\right)}^2-1=0\cr {\left(x+y\right)}^2-3^2=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&x-y=1\,{\mbox{ and }}\, x+y=3\,{\mbox{ or }}\, x-y=-1\,{\mbox{ and }}\, x+y=3\,{\mbox{ or }}\, x-y=1\,{\mbox{ and }}\, x+y=-3\,{\mbox{ or }}\, x-y=-1\,{\mbox{ and }}\, x+y=-3& \cr \color{green}{\Leftrightarrow}&x=1\,{\mbox{ and }}\, y=2\,{\mbox{ or }}\, x=2\,{\mbox{ and }}\, y=1\,{\mbox{ or }}\, x=-2\,{\mbox{ and }}\, y=-1\,{\mbox{ or }}\, x=-1\,{\mbox{ and }}\, y=-2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [4*x^2+7*x*y+4*y^2=4 and y=x-4, 4*x^2+7*x*(x-4)+4*(x-4)^2-4=0 and y=x-4, 15*x^2-60*x+60=0 and y=x-4, (x-2)^2=0 and y=x-4, x=2 and y=x-4, x=2 and y=-2] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}4\cdot x^2+7\cdot x\cdot y+4\cdot y^2=4\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}4\cdot x^2+7\cdot x\cdot \left(x-4\right)+4\cdot {\left(x-4\right)}^2-4=0\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}15\cdot x^2-60\cdot x+60=0\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}{\left(x-2\right)}^2=0\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x=2\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x=2\cr y=-2\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a^2=b and a^2=1, b=a^2 and (a=1 or a=-1), (b=1 and a=1) or (b=1 and a=-1)] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}a^2=b\cr a^2=1\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a=1\,{\mbox{ or }}\, a=-1\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&b=1\,{\mbox{ and }}\, a=1\,{\mbox{ or }}\, b=1\,{\mbox{ and }}\, a=-1& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [a^2=b and x=1, b=a^2 and x=1] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}a^2=b\cr x=1\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr x=1\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR]
Equiv Pass [a^2=b and b^2=a, b=a^2 and a^4=a, b=a^2 and a^4-a=0, b=a^2 and a*(a-1)*(a^2+a+1)=0, b=a^2 and (a=0 or a=1 or a^2+a+1=0), (b=0 and a=0) or (b=1 and a=1)] [] [assumereal] 1 1
\[\begin{array}{lll}\color{blue}{(\mathbb{R})}&\left\{\begin{array}{l}a^2=b\cr b^2=a\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a^4=a\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a^4-a=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a\cdot \left(a-1\right)\cdot \left(a^2+a+1\right)=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a=0\,{\mbox{ or }}\, a=1\,{\mbox{ or }}\, a^2+a+1=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&b=0\,{\mbox{ and }}\, a=0\,{\mbox{ or }}\, b=1\,{\mbox{ and }}\, a=1& \cr \end{array}\]
[ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [2*x^3-9*x^2+10*x-3,stacklet(x,1),2*1^3-9*1^2+10*1-3,stackeq(0),"So",2*x^3-9*x^2+10*x-3,stackeq((x-1)*(2*x^2-7*x+3)),stackeq((x-1)*(2*x-1)*(x-3))] [] 0 0
\[\begin{array}{lll} &2\cdot x^3-9\cdot x^2+10\cdot x-3& \cr &\mbox{Let }x = 1& \cr \color{green}{\Leftrightarrow}&2\cdot 1^3-9\cdot 1^2+10\cdot 1-3& \cr \color{green}{\checkmark}&=0& \cr &\mbox{So}& \cr &2\cdot x^3-9\cdot x^2+10\cdot x-3& \cr \color{green}{\checkmark}&=\left(x-1\right)\cdot \left(2\cdot x^2-7\cdot x+3\right)& \cr \color{green}{\checkmark}&=\left(x-1\right)\cdot \left(2\cdot x-1\right)\cdot \left(x-3\right)& \cr \end{array}\]
[EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, CHECKMARK, EMPTYCHAR, EMPTYCHAR, CHECKMARK, CHECKMARK]
Equiv Pass [2*x^2+x>=6, 2*x^2+x-6>=0, (2*x-3)*(x+2)>= 0,((2*x-3)>=0 and (x+2)>=0) or ((2*x-3)<=0 and (x+2)<=0), (x>=3/2 and x>=-2) or (x<=3/2 and x<=-2), x>=3/2 or x <=-2] [] 1 1
\[\begin{array}{lll} &2\cdot x^2+x\geq 6& \cr \color{green}{\Leftrightarrow}&2\cdot x^2+x-6\geq 0& \cr \color{green}{\Leftrightarrow}&\left(2\cdot x-3\right)\cdot \left(x+2\right)\geq 0& \cr \color{green}{\Leftrightarrow}&2\cdot x-3\geq 0\,{\mbox{ and }}\, x+2\geq 0\,{\mbox{ or }}\, 2\cdot x-3\leq 0\,{\mbox{ and }}\, x+2\leq 0& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ and }}\, x\geq -2\,{\mbox{ or }}\, x\leq \frac{3}{2}\,{\mbox{ and }}\, x\leq -2& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ or }}\, x\leq -2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [2*x^2+x>=6, 2*x^2+x-6>=0, (2*x-3)*(x+2)>= 0,((2*x-3)>=0 and (x+2)>=0) or ((2*x-3)<=0 and (x+2)<=0), (x>=3/2 and x>=-2) or (x<=3/2 and x<=-2), x>=3/2 or x <=-2] [] 1 1
\[\begin{array}{lll} &2\cdot x^2+x\geq 6& \cr \color{green}{\Leftrightarrow}&2\cdot x^2+x-6\geq 0& \cr \color{green}{\Leftrightarrow}&\left(2\cdot x-3\right)\cdot \left(x+2\right)\geq 0& \cr \color{green}{\Leftrightarrow}&2\cdot x-3\geq 0\,{\mbox{ and }}\, x+2\geq 0\,{\mbox{ or }}\, 2\cdot x-3\leq 0\,{\mbox{ and }}\, x+2\leq 0& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ and }}\, x\geq -2\,{\mbox{ or }}\, x\leq \frac{3}{2}\,{\mbox{ and }}\, x\leq -2& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ or }}\, x\leq -2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [2*x^2+x>=6, 2*x^2+x-6>=0, (2*x-3)*(x+2)>= 0,((2*x-3)>=0 and (x+2)>=0) or ((2*x-3)<=0 and (x+2)<=0), (x>=3/2 and x>=-2) or (x<=3/2 and x<=-2), x>=3/2 or x <=2] [] 0 0
\[\begin{array}{lll} &2\cdot x^2+x\geq 6& \cr \color{green}{\Leftrightarrow}&2\cdot x^2+x-6\geq 0& \cr \color{green}{\Leftrightarrow}&\left(2\cdot x-3\right)\cdot \left(x+2\right)\geq 0& \cr \color{green}{\Leftrightarrow}&2\cdot x-3\geq 0\,{\mbox{ and }}\, x+2\geq 0\,{\mbox{ or }}\, 2\cdot x-3\leq 0\,{\mbox{ and }}\, x+2\leq 0& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ and }}\, x\geq -2\,{\mbox{ or }}\, x\leq \frac{3}{2}\,{\mbox{ and }}\, x\leq -2& \cr \color{red}{?}&x\geq \frac{3}{2}\,{\mbox{ or }}\, x\leq 2& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR]
Equiv Pass [x^2>=9 and x>3, x^2-9>=0 and x>3, (x>=3 or x<=-3) and x>3, x>3] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}x^2\geq 9\cr x > 3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x^2-9\geq 0\cr x > 3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x\geq 3\,{\mbox{ or }}\, x\leq -3\cr x > 3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&x > 3& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [-x^2+a*x+a-3<0, a-3<x^2-a*x, a-3<(x-a/2)^2-a^2/4, a^2/4+a-3<(x-a/2)^2, a^2+4*a-12<4*(x-a/2)^2, (a-2)*(a+6)<4*(x-a/2)^2, "This inequality is required to be true for all x.", "So it must be true when the right hand side takes its minimum value.", "This happens for x=a/2.", (a-2)*(a+6)<0, ((a-2)<0 and (a+6)>0) or ((a-2)>0 and (a+6)<0), (a<2 and a>-6) or (a>2 and a<-6), (-6<a and a<2) or false, (-6<a and a<2)] [] 0 0
\[\begin{array}{lll} &-x^2+a\cdot x+a-3 < 0& \cr \color{green}{\Leftrightarrow}&a-3 < x^2-a\cdot x& \cr \color{green}{\Leftrightarrow}&a-3 < {\left(x-\frac{a}{2}\right)}^2-\frac{a^2}{4}& \cr \color{green}{\Leftrightarrow}&\frac{a^2}{4}+a-3 < {\left(x-\frac{a}{2}\right)}^2& \cr \color{green}{\Leftrightarrow}&a^2+4\cdot a-12 < 4\cdot {\left(x-\frac{a}{2}\right)}^2& \cr \color{green}{\Leftrightarrow}&\left(a-2\right)\cdot \left(a+6\right) < 4\cdot {\left(x-\frac{a}{2}\right)}^2& \cr &\mbox{This inequality is required to be true for all x.}& \cr &\mbox{So it must be true when the right hand side takes its minimum value.}& \cr &\mbox{This happens for x=a/2.}& \cr &\left(a-2\right)\cdot \left(a+6\right) < 0& \cr \color{green}{\Leftrightarrow}&a-2 < 0\,{\mbox{ and }}\, a+6 > 0\,{\mbox{ or }}\, a-2 > 0\,{\mbox{ and }}\, a+6 < 0& \cr \color{green}{\Leftrightarrow}&a < 2\,{\mbox{ and }}\, a > -6\,{\mbox{ or }}\, a > 2\,{\mbox{ and }}\, a < -6& \cr \color{green}{\Leftrightarrow}&-6 < a\,{\mbox{ and }}\, a < 2\,{\mbox{ or }}\, \mathbf{False}& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}-6 < a\cr a < 2\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x-2>0 and x*(x-2)<15,x>2 and x^2-2*x-15<0,x>2 and (x-5)*(x+3)<0,x>2 and ((x<5 and x>-3) or (x>5 and x<-3)),x>2 and (x<5 and x>-3),x>2 and x<5] [] 1 1
\[\begin{array}{lll} &\left\{\begin{array}{l}x-2 > 0\cr x\cdot \left(x-2\right) < 15\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x > 2\cr x^2-2\cdot x-15 < 0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x > 2\cr \left(x-5\right)\cdot \left(x+3\right) < 0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x > 2\cr x < 5\,{\mbox{ and }}\, x > -3\,{\mbox{ or }}\, x > 5\,{\mbox{ and }}\, x < -3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x > 2\cr x < 5\,{\mbox{ and }}\, x > -3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x > 2\cr x < 5\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x-2>0 and x*(x-2)<15,x>2 and x^2-2*x-15<0,x>2 and (x-5)*(x+3)<0,x>2 and ((x<5 and x>-3) or (x>5 and x<-3)),x>7 and (x<5 and x>-3),x>2 and x<5] [] 0 0
\[\begin{array}{lll} &\left\{\begin{array}{l}x-2 > 0\cr x\cdot \left(x-2\right) < 15\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x > 2\cr x^2-2\cdot x-15 < 0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x > 2\cr \left(x-5\right)\cdot \left(x+3\right) < 0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x > 2\cr x < 5\,{\mbox{ and }}\, x > -3\,{\mbox{ or }}\, x > 5\,{\mbox{ and }}\, x < -3\cr \end{array}\right.& \cr \color{red}{?}&\left\{\begin{array}{l}x > 7\cr x < 5\,{\mbox{ and }}\, x > -3\cr \end{array}\right.& \cr \color{red}{?}&\left\{\begin{array}{l}x > 2\cr x < 5\cr \end{array}\right.& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR,QMCHAR]
Equiv Pass [x^2 + (a-2)*x + a = 0,(x + (a-2)/2)^2 -((a-2)/2)^2 + a = 0,(x + (a-2)/2)^2 =(a-2)^2/4 - a,"This has real roots iff",(a-2)^2/4-a >=0,a^2-4*a+4-4*a >=0,a^2-8*a+4>=0,(a-4)^2-16+4>=0,(a-4)^2>=12,a-4>=sqrt(12) or a-4<= -sqrt(12),"Ignoring the negative solution.",a>=sqrt(12)+4,"Using external domain information that a is an integer.",a>=8] [] 0 0
\[\begin{array}{lll} &x^2+\left(a-2\right)\cdot x+a=0& \cr \color{green}{\Leftrightarrow}&{\left(x+\frac{a-2}{2}\right)}^2-{\left(\frac{a-2}{2}\right)}^2+a=0& \cr \color{green}{\Leftrightarrow}&{\left(x+\frac{a-2}{2}\right)}^2=\frac{{\left(a-2\right)}^2}{4}-a& \cr &\mbox{This has real roots iff}& \cr &\frac{{\left(a-2\right)}^2}{4}-a\geq 0& \cr \color{green}{\Leftrightarrow}&a^2-4\cdot a+4-4\cdot a\geq 0& \cr \color{green}{\Leftrightarrow}&a^2-8\cdot a+4\geq 0& \cr \color{green}{\Leftrightarrow}&{\left(a-4\right)}^2-16+4\geq 0& \cr \color{green}{\Leftrightarrow}&{\left(a-4\right)}^2\geq 12& \cr \color{green}{\Leftrightarrow}&a-4\geq \sqrt{12}\,{\mbox{ or }}\, a-4\leq -\sqrt{12}& \cr &\mbox{Ignoring the negative solution.}& \cr &a\geq \sqrt{12}+4& \cr &\mbox{Using external domain information that a is an integer.}& \cr &a\geq 8& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR]
Equiv Pass ["Set P(n) be the statement that",sum(k^2,k,1,n) = n*(n+1)*(2*n+1)/6, "Then P(1) is the statement", 1^2 = 1*(1+1)*(2*1+1)/6, 1 = 1, "So P(1) holds. Now assume P(n) is true.",sum(k^2,k,1,n) = n*(n+1)*(2*n+1)/6,sum(k^2,k,1,n) +(n+1)^2= n*(n+1)*(2*n+1)/6 +(n+1)^2,sum(k^2,k,1,n+1)= (n+1)*(n*(2*n+1) +6*(n+1))/6,sum(k^2,k,1,n+1)= (n+1)*(2*n^2+7*n+6)/6,sum(k^2,k,1,n+1)= (n+1)*(n+1+1)*(2*(n+1)+1)/6] [] 0 0
\[\begin{array}{lll} &\mbox{Set P(n) be the statement that}& \cr &\sum_{k=1}^{n}{k^2}=\frac{n\cdot \left(n+1\right)\cdot \left(2\cdot n+1\right)}{6}& \cr &\mbox{Then P(1) is the statement}& \cr &1^2=\frac{1\cdot \left(1+1\right)\cdot \left(2\cdot 1+1\right)}{6}& \cr \color{green}{\Leftrightarrow}&1=1& \cr &\mbox{So P(1) holds. Now assume P(n) is true.}& \cr &\sum_{k=1}^{n}{k^2}=\frac{n\cdot \left(n+1\right)\cdot \left(2\cdot n+1\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n}{k^2}+{\left(n+1\right)}^2=\frac{n\cdot \left(n+1\right)\cdot \left(2\cdot n+1\right)}{6}+{\left(n+1\right)}^2& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(n\cdot \left(2\cdot n+1\right)+6\cdot \left(n+1\right)\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(2\cdot n^2+7\cdot n+6\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(n+1+1\right)\cdot \left(2\cdot \left(n+1\right)+1\right)}{6}& \cr \end{array}\]
[EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [(n+1)^2+sum(k^2,k,1,n) = (n+1)^2+(n*(n+1)*(2*n+1))/6, sum(k^2,k,1,n+1) = ((n+1)*(n*(2*n+1)+6*(n+1)))/6, sum(k^2,k,1,n+1) = ((n+1)*(2*n^2+7*n+6))/6, sum(k^2,k,1,n+1) = ((n+1)*(n+2)*(2*(n+1)+1))/6] [] 1 1
\[\begin{array}{lll} &{\left(n+1\right)}^2+\sum_{k=1}^{n}{k^2}={\left(n+1\right)}^2+\frac{n\cdot \left(n+1\right)\cdot \left(2\cdot n+1\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(n\cdot \left(2\cdot n+1\right)+6\cdot \left(n+1\right)\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(2\cdot n^2+7\cdot n+6\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(n+2\right)\cdot \left(2\cdot \left(n+1\right)+1\right)}{6}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [conjugate(a)*conjugate(b),stacklet(a,x+i*y),stacklet(b,r+i*s),stackeq(conjugate(x+i*y)*conjugate(r+i*s)),stackeq((x-i*y)*(r-i*s)),stackeq((x*r-y*s)-i*(y*r+x*s)),stackeq(conjugate((x*r-y*s)+i*(y*r+x*s))),stackeq(conjugate((x+i*y)*(r+i*s))),stacklet(x+i*y,a),stacklet(r+i*s,b),stackeq(conjugate(a*b))] [] 1 1
\[\begin{array}{lll} &a^\star\cdot b^\star& \cr &\mbox{Let }a = x+\mathrm{i}\cdot y& \cr &\mbox{Let }b = r+\mathrm{i}\cdot s& \cr \color{green}{\checkmark}&=\left(x+\mathrm{i}\cdot y\right)^\star\cdot \left(r+\mathrm{i}\cdot s\right)^\star& \cr \color{green}{\checkmark}&=\left(x-\mathrm{i}\cdot y\right)\cdot \left(r-\mathrm{i}\cdot s\right)& \cr \color{green}{\checkmark}&=x\cdot r-y\cdot s-\mathrm{i}\cdot \left(y\cdot r+x\cdot s\right)& \cr \color{green}{\checkmark}&=\left(x\cdot r-y\cdot s+\mathrm{i}\cdot \left(y\cdot r+x\cdot s\right)\right)^\star& \cr \color{green}{\checkmark}&=\left(\left(x+\mathrm{i}\cdot y\right)\cdot \left(r+\mathrm{i}\cdot s\right)\right)^\star& \cr &\mbox{Let }x+\mathrm{i}\cdot y = a& \cr &\mbox{Let }r+\mathrm{i}\cdot s = b& \cr \color{green}{\checkmark}&=\left(a\cdot b\right)^\star& \cr \end{array}\]
[EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, EMPTYCHAR, EMPTYCHAR, CHECKMARK]
Equiv Pass [nounint(x*e^x,x,-inf,0),nounlimit(nounint(x*e^x,x,t,0),t,-inf),nounlimit(e^t-t*e^t-1,t,-inf),nounlimit(e^t,t,-inf)+nounlimit(-t*e^t,t,-inf)+nounlimit(-1,t,-inf),-1] [] 1 1
\[\begin{array}{lll} &\int_{-\infty }^{0}{x\cdot e^{x}\;\mathrm{d}x}& \cr \color{green}{\Leftrightarrow}&\lim_{t\rightarrow -\infty }{\int_{t}^{0}{x\cdot e^{x}\;\mathrm{d}x}}& \cr \color{green}{\Leftrightarrow}&\lim_{t\rightarrow -\infty }{e^{t}-t\cdot e^{t}-1}& \cr \color{green}{\Leftrightarrow}&\lim_{t\rightarrow -\infty }{e^{t}}+\lim_{t\rightarrow -\infty }{\left(-t\right)\cdot e^{t}}+\lim_{t\rightarrow -\infty }{-1}& \cr \color{green}{\Leftrightarrow}&-1& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [noundiff(x^2,x),stackeq(nounlimit(((x+h)^2-x^2)/h,h,0)),stackeq(nounlimit(2*x+h,h,0)),stackeq(2*x)] [] 1 1
\[\begin{array}{lll} &\frac{\mathrm{d}}{\mathrm{d} x} x^2& \cr \color{green}{\checkmark}&=\lim_{h\rightarrow 0}{\frac{{\left(x+h\right)}^2-x^2}{h}}& \cr \color{green}{\checkmark}&=\lim_{h\rightarrow 0}{2\cdot x+h}& \cr \color{green}{\checkmark}&=2\cdot x& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK]
Equiv Pass [-12+3*noundiff(y(x),x)+8-8*noundiff(y(x),x)=0,-5*noundiff(y(x),x)=4,noundiff(y(x),x)=-4/5] [] [calculus] 1 1
\[\begin{array}{lll} &-12+3\cdot \left(\frac{\mathrm{d}}{\mathrm{d} x} y\left(x\right)\right)+8-8\cdot \left(\frac{\mathrm{d}}{\mathrm{d} x} y\left(x\right)\right)=0& \cr \color{green}{\Leftrightarrow}&-5\cdot \left(\frac{\mathrm{d}}{\mathrm{d} x} y\left(x\right)\right)=4& \cr \color{green}{\Leftrightarrow}&\left(\frac{\mathrm{d}}{\mathrm{d} x} y\left(x\right)\right)=\frac{-4}{5}& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR]
Equiv Pass [x^2+1,x^3/3+x,x^2+1,x^3/3+x+c] [] [calculus] 1 1
\[\begin{array}{lll} &x^2+1& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{x^3}{3}+x& \cr \color{blue}{\frac{\mathrm{d}}{\mathrm{d}x}\ldots}&x^2+1& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{x^3}{3}+x+c& \cr \end{array}\]
[EMPTYCHAR,INTCHAR(x),DIFFCHAR(x),INTCHAR(x)]
Equiv Pass [3*x^(3/2)-2/x,(9*sqrt(x))/2+2/x^2,3*x^(3/2)-2/x+c] [] [calculus] 1 1
\[\begin{array}{lll} &3\cdot x^{\frac{3}{2}}-\frac{2}{x}&{\color{blue}{{x \not\in {\left \{0 \right \}}}}}\cr \color{blue}{\frac{\mathrm{d}}{\mathrm{d}x}\ldots}&\frac{9\cdot \sqrt{x}}{2}+\frac{2}{x^2}&{\color{blue}{{x \in {\left( 0,\, \infty \right)}}}}\cr \color{blue}{\int\ldots\mathrm{d}x}&3\cdot x^{\frac{3}{2}}-\frac{2}{x}+c& \cr \end{array}\]
[EMPTYCHAR,DIFFCHAR(x),INTCHAR(x)]
Equiv Pass [x^2+1,stackeq(x^3/3+x),stackeq(x^2+1),stackeq(x^3/3+x+c)] [] [calculus] 0 0
\[\begin{array}{lll} &x^2+1& \cr \color{red}{?}&=\frac{x^3}{3}+x& \cr \color{red}{?}&=x^2+1& \cr \color{red}{?}&=\frac{x^3}{3}+x+c& \cr \end{array}\]
[EMPTYCHAR,QMCHAR,QMCHAR,QMCHAR]
Equiv Pass [diff(x^2*sin(x),x),stackeq(x^2*diff(sin(x),x)+diff(x^2,x)*sin(x)),stackeq(x^2*cos(x)+2*x*sin(x))] [] [calculus] 1 1
\[\begin{array}{lll} &\cos \left( x \right)\cdot x^2+2\cdot x\cdot \sin \left( x \right)& \cr \color{green}{\checkmark}&=x^2\cdot \cos \left( x \right)+2\cdot x\cdot \sin \left( x \right)& \cr \color{green}{\checkmark}&=x^2\cdot \cos \left( x \right)+2\cdot x\cdot \sin \left( x \right)& \cr \end{array}\]
[EMPTYCHAR, CHECKMARK, CHECKMARK]
Equiv Pass [y(x)*cos(x)+y(x)^2 = 6*x,cos(x)*diff(y(x),x)+2*y(x)*diff(y(x),x)-y(x)*sin(x) = 6,(cos(x)+2*y(x))*diff(y(x),x) = y(x)*sin(x)+6,diff(y(x),x) = (y(x)*sin(x)+6)/(cos(x)+2*y(x))] [] [calculus] 1 1
\[\begin{array}{lll} &y\left(x\right)\cdot \cos \left( x \right)+y^2\left(x\right)=6\cdot x& \cr \color{blue}{\frac{\mathrm{d}}{\mathrm{d}x}\ldots}&\cos \left( x \right)\cdot \left(\frac{\mathrm{d}}{\mathrm{d} x} y\left(x\right)\right)+2\cdot y\left(x\right)\cdot \left(\frac{\mathrm{d}}{\mathrm{d} x} y\left(x\right)\right)+\left(-y\left(x\right)\right)\cdot \sin \left( x \right)=6& \cr \color{green}{\Leftrightarrow}&\left(\cos \left( x \right)+2\cdot y\left(x\right)\right)\cdot \left(\frac{\mathrm{d}}{\mathrm{d} x} y\left(x\right)\right)=y\left(x\right)\cdot \sin \left( x \right)+6& \cr \color{green}{\Leftrightarrow}&\left(\frac{\mathrm{d}}{\mathrm{d} x} y\left(x\right)\right)=\frac{y\left(x\right)\cdot \sin \left( x \right)+6}{\cos \left( x \right)+2\cdot y\left(x\right)}& \cr \end{array}\]
[EMPTYCHAR,DIFFCHAR(x), EQUIVCHAR, EQUIVCHAR]
Equiv Pass [nounint(s^2+1,s),stackeq(s^3/3+s+c)] [] [calculus] 1 1
\[\begin{array}{lll} &\int {s^2+1}{\;\mathrm{d}s}& \cr \color{blue}{\int\ldots\mathrm{d}s}&=\frac{s^3}{3}+s+c& \cr \end{array}\]
[EMPTYCHAR,INTCHAR(s)]
Equiv Pass [nounint(x^3*log(x),x),x^4/4*log(x)-1/4*nounint(x^3,x),x^4/4*log(x)-x^4/16] [] [calculus] 0 0
\[\begin{array}{lll} &\int {x^3\cdot \ln \left( x \right)}{\;\mathrm{d}x}& \cr \color{green}{\Leftrightarrow}&\frac{x^4}{4}\cdot \ln \left( x \right)-\frac{1}{4}\cdot \int {x^3}{\;\mathrm{d}x}& \cr \color{red}{\cdots +c\quad ?}&\frac{x^4}{4}\cdot \ln \left( x \right)-\frac{x^4}{16}&{\color{blue}{{x \in {\left( 0,\, \infty \right)}}}}\cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR,PLUSC]
Equiv Pass [nounint(x^3*log(x),x),x^4/4*log(x)-1/4*nounint(x^3,x),x^4/4*log(x)-x^4/16+c] [] [calculus] 1 1
\[\begin{array}{lll} &\int {x^3\cdot \ln \left( x \right)}{\;\mathrm{d}x}& \cr \color{green}{\Leftrightarrow}&\frac{x^4}{4}\cdot \ln \left( x \right)-\frac{1}{4}\cdot \int {x^3}{\;\mathrm{d}x}& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{x^4}{4}\cdot \ln \left( x \right)-\frac{x^4}{16}+c& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR,INTCHAR(x)]
Equiv Pass [noundiff(y,x)-2/x*y=x^3*sin(3*x),1/x^2*noundiff(y,x)-2/x^3*y=x*sin(3*x),noundiff(y/x^2,x)=x*sin(3*x),y/x^2 = nounint(x*sin(3*x),x),y/x^2=(sin(3*x)-3*x*cos(3*x))/9+c] [] [calculus] 1 1
\[\begin{array}{lll} &\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{2}{x}\cdot y=x^3\cdot \sin \left( 3\cdot x \right)& \cr \color{green}{\Leftrightarrow}&\frac{1}{x^2}\cdot \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)-\frac{2}{x^3}\cdot y=x\cdot \sin \left( 3\cdot x \right)& \cr \color{green}{\Leftrightarrow}&\left(\frac{\mathrm{d}}{\mathrm{d} x} \frac{y}{x^2}\right)=x\cdot \sin \left( 3\cdot x \right)& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{y}{x^2}=\int {x\cdot \sin \left( 3\cdot x \right)}{\;\mathrm{d}x}& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{y}{x^2}=\frac{\sin \left( 3\cdot x \right)-3\cdot x\cdot \cos \left( 3\cdot x \right)}{9}+c& \cr \end{array}\]
[EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,INTCHAR(x),INTCHAR(x)]

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