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.

## AlgEquiv

Test
?
Student response
Opt
Mark
CAS errors
Feedback
AlgEquiv
1/0
1
AlgEquiv
1
1/0
AlgEquiv
(x-1)^2
AlgEquiv
x^2
AlgEquiv
x-1)^2
(x-1)^2
See docs on subscripts and different atoms.
AlgEquiv
x1
x_1
0
AlgEquiv
x_1
x[1]
0
AlgEquiv
x[1]
x1
0
Predicates
AlgEquiv
integerp(3)
true
1 ATLogic_True.
AlgEquiv
integerp(3.1)
true
0
AlgEquiv
integerp(3)
false
0
AlgEquiv
integerp(3)
true
1 ATLogic_True.
AlgEquiv
lowesttermsp(x^2/x)
true
1 ATLogic_True.
AlgEquiv
lowesttermsp(-y/-x)
true
1 ATLogic_True.
AlgEquiv
lowesttermsp((x^2-1)/(x-1))
true
0
AlgEquiv
lowesttermsp((x^2-1)/(x+2))
true
1 ATLogic_True.
Case sensitivity
AlgEquiv
X
x
0 ATAlgEquiv_WrongCase.
AlgEquiv
exdowncase(X)
x
1
AlgEquiv
exdowncase((X-1)^2)
x^2-2*x+1
1
Permutations of variables (To do: a dedicated answer test with feedback)
AlgEquiv
Y=1+X
y=1+x
0 ATEquation_default
AlgEquiv
v+w+x+y+z
a+b+c+A+B
0
Numbers
AlgEquiv
4^(-1/2)
1/2
1
AlgEquiv
4^(1/2)
sqrt(4)
1
Mix of floats and rational numbers
AlgEquiv
0.5
1/2
1
AlgEquiv
0.33
1/3
0
AlgEquiv
452
4.52*10^2
0
AlgEquiv
5.1e-2
51/1000
1
AlgEquiv
0.333333333333333
1/3
0
AlgEquiv
(0.5+x)*2
2*x+1
1
Complex numbers
AlgEquiv
sqrt(-1)
%i
1
AlgEquiv
%i
e^(i*pi/2)
1
AlgEquiv
(4*sqrt(3)*%i+4)^(1/5)
8^(1/5)*(cos(%pi/15)+%i*sin(%p
i/15))
1
AlgEquiv
(4*sqrt(3)*%i+4)^(1/5)
rectform((4*sqrt(3)*%i+4)^(1/5
))
1
AlgEquiv
(4*sqrt(3)*%i+4)^(1/5)
polarform((4*sqrt(3)*%i+4)^(1/
5))
1
AlgEquiv
%i/sqrt(x)
sqrt(-1/x)
1
Infinity
AlgEquiv
inf
inf
1
AlgEquiv
inf
-inf
0
AlgEquiv
2*inf
inf
0
AlgEquiv
0*inf
0
1
Powers and roots
AlgEquiv
x^(1/2)
sqrt(x)
1
AlgEquiv
x
sqrt(x^2)
0
AlgEquiv
abs(x)
sqrt(x^2)
1
AlgEquiv
1/abs(x)^(1/3)
(abs(x)^(1/3)/abs(x))^(1/2)
1
AlgEquiv
sqrt((x-3)*(x-5))
sqrt(x-3)*sqrt(x-5)
0
AlgEquiv
1/sqrt(x)
sqrt(1/x)
1
AlgEquiv
x-1
(x^2-1)/(x+1)
1
AlgEquiv
2^((1/5.1)*t)
2^((1/5.1)*t)
1
AlgEquiv
2^((1/5.1)*t)
2^(0.196078431373*t)
0
AlgEquiv
a^b*a^c
a^(b+c)
1
AlgEquiv
(a^b)^c
a^(b*c)
0
AlgEquiv
(assume(a>0),(a^b)^c)
a^(b*c)
1
AlgEquiv
(assume(x>2),6*((x-2)^2)^k)
6*(x-2)^(2*k)
1
AlgEquiv
signum(-3)
-1
1
AlgEquiv
6*((x-2)^3)^k
6*(x-2)^(3*k)
1
AlgEquiv
(4*sqrt(3)*%i+4)^(1/5)
6^(1/5)*cos(%pi/15)-6^(1/5)*%i
*sin(%pi/15)
0
AlgEquiv
2+2*sqrt(3+x)
2+sqrt(12+4*x)
1
AlgEquiv
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
Expressions with subscripts
AlgEquiv
a1
a_1
0
AlgEquiv
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
AlgEquiv
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
AlgEquiv
sqrt(k/m)*sqrt(m/k)
1
1
AlgEquiv
(2*pi)/(k/m)^(1/2)
(2*pi)/(k/m)^(1/2)
1
AlgEquiv
(2*pi)*(m/k)^(1/2)
(2*pi)/(k/m)^(1/2)
1
AlgEquiv
sqrt(2*x/10+1)
sqrt((2*x+10)/10)
1
AlgEquiv
((x+3)^2*(x+3))^(1/3)
((x+3)*(x^2+6*x+9))^(1/3)
1
Need to factor internally.
AlgEquiv
((x+3)^2*(x+3))^(1/3)
((x+3)*(x^2+6*x+9))^(1/3)
1
Polynomials and rational function
AlgEquiv
(x-1)^2
x^2-2*x+1
1
AlgEquiv
(x-1)*(x^2+x+1)
x^3-1
1
AlgEquiv
(x-1)^(-2)
1/(x^2-2*x+1)
1
AlgEquiv
1/(4*x-(%pi+sqrt(2)))
1/(x+1)
0
AlgEquiv
(x-a)^6000
(x-a)^6000
1
AlgEquiv
(a-x)^6000
(x-a)^6000
1
AlgEquiv
(4*a-x)^6000
(x-4*a)^6000
1
AlgEquiv
(x-a)^6000
(x-a)^5999
0
AlgEquiv
(k+8)/(k^2+4*k-12)
(k+8)/(k^2+4*k-12)
1
AlgEquiv
(k+7)/(k^2+4*k-12)
(k+8)/(k^2+4*k-12)
0
AlgEquiv
-(2*k+6)/(k^2+4*k-12)
-(2*k+6)/(k^2+4*k-12)
1
AlgEquiv
1/n-1/(n+1)
1/(n*(n+1))
1
AlgEquiv
0.5*x^2+3*x-1
x^2/2+3*x-1
1
AlgEquiv
14336000000*x^13+250265600000*
x^12+1862860800000*x^11+762392
5760000*x^10+18290677760000*x^
9+24744757985280*x^8+145672123
51488*x^7-3267871272960*x^6-64
08053107200*x^5+670406720000*x
^4+1179708800000*x^3-429244800
000*x^2+56696000000*x-26800000
00
512*(2*x+5)^7*(5*x-1)^5*(70*x+
67)
1
AlgEquiv
14336000000*x^13+250265600000*
x^12+1862860800000*x^11+762392
5760000*x^10+18290677760000*x^
9+24744757985280*x^8+145672123
51488*x^7-3267871272960*x^6-64
08053107200*x^5+670406720000*x
^4+1179708800000*x^3-429244800
000*x^2+56696000000*x-26800000
01
512*(2*x+5)^7*(5*x-1)^5*(70*x+
67)
0
AlgEquiv
14336000000*x^13
512*(2*x+5)^7*(5*x-1)^5*(70*x+
67)
0
Trig functions
AlgEquiv
cos(x)
cos(-x)
1
AlgEquiv
cos(x)^2+sin(x)^2
1
1
AlgEquiv
cos(x+y)
cos(x)*cos(y)-sin(x)*sin(y)
1
AlgEquiv
cos(x+y)
cos(x)*cos(y)+sin(x)*sin(y)
0
AlgEquiv
cos(x#pm#y)
cos(x)*cos(y)-(#pm#sin(x)*sin(
y))
1 ATLogic_True.
AlgEquiv
sin(x#pm#y)
sin(x)*cos(y)#pm#cos(x)*sin(y)
1 ATLogic_True.
AlgEquiv
sin(x#pm#y)
cos(x)*sin(y)#pm#sin(x)*cos(y)
0
AlgEquiv
2*cos(x)^2-1
cos(2*x)
1
AlgEquiv
1.0*cos(1200*%pi*x)
cos(1200*%pi*x)
1
AlgEquiv
diff(tan(10*x)^2,x)
cos(6*x)
0
AlgEquiv
exp(%i*%pi)
-1
1
AlgEquiv
2*cos(2*x)+x+1
-sin(x)^2+3*cos(x)^2+x
1
AlgEquiv
(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
AlgEquiv
1+cosec(3*x)
1+csc(3*x)
1
AlgEquiv
1/(1+exp(-2*x))
tanh(x)/2+1/2
1
AlgEquiv
1+cosech(3*x)
1+csch(3*x)
1
AlgEquiv
-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
AlgEquiv
-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
AlgEquiv
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
AlgEquiv
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
AlgEquiv
-(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
AlgEquiv
sin((2*%pi*n-%pi)/2)
-cos(n*%pi)
1
AlgEquiv
sin(x/2)/(1+tan(x)*tan(x/2))
sin(x/2)*cos(x)
1
AlgEquiv
(declare(n,integer),trigrat(si
n((2*%pi*n-%pi)/2)))
-(-1)^n
1
AlgEquiv !
cot(%pi/20)+cot(%pi/24)-cot(%p
i/10)
sqrt(1)+sqrt(2)+sqrt(3)+sqrt(4
)+sqrt(5)+sqrt(6)
-3
AlgEquiv
trigeval(cot(%pi/20)+cot(%pi/2
4)-cot(%pi/10))
sqrt(1)+sqrt(2)+sqrt(3)+sqrt(4
)+sqrt(5)+sqrt(6)
1
AlgEquiv !
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]
-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
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
AlgEquiv
1+x
taylor(1/(1-x),x,0,1)
1
AlgEquiv
1
taylor(1/(1-x),x,0,1)
0
Logarithms
AlgEquiv
log(a^2*b)
2*log(a)+log(b)
1
AlgEquiv
(2*log(2*x)+x)/(2*x)
(log(2*x)+2)/(2*sqrt(x))
0
AlgEquiv
log(abs((x^2-9)))
log(abs(x-3))+log(abs(x+3))
0
AlgEquiv
lg(10^x)
x
1
AlgEquiv
lg(3^x,3)
x
1
AlgEquiv
lg(a^x,a)
x
1
AlgEquiv
1+lg(27,3)
4
1
AlgEquiv
1+lg(27,3)
3
0
AlgEquiv
lg(1/8,2)
-3
1
AlgEquiv
lg(root(x,n))
lg(x,10)/n
1
AlgEquiv
log(root(x,n))
lg(x,10)/n
0
AlgEquiv
x^log(y)
y^log(x)
1
Hyperbolic trig
AlgEquiv
e^1-e^(-1)
2*sinh(1)
1
Lists
AlgEquiv
x
[1,2,3]
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
[1,2]
[1,2,3]
0 Your list should have $$3$$ elements, but it actually has $$2$$. ATList_wronglen.
AlgEquiv
[1,2,4]
[1,2,3]
0 The entries underlined in red below are those that are incorrect. $\left[ 1 , 2 , {\color{red}{\underline{4}}} \right]$ (ATList_wrongentries 3).
AlgEquiv
[1,x>2]
[1,2<x]
1
AlgEquiv
[1,2,[2-x<0,{1,2,2,2, 1,3}]
]
[1,2,[2-x<0,{1,2}]]
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
[(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
AlgEquiv
[1,2]
ntuple(1,2)
0 Your answer should be an expression, not an equation, inequality, list, set or matrix. ATAlgEquiv_SA_not_expression.
Rounding of floats
AlgEquiv
round(0.5)
0.0
1
AlgEquiv
round(1.5)
2.0
1
AlgEquiv
round(2.5)
2.0
1
AlgEquiv
round(12.5)
12.0
1
AlgEquiv
significantfigures(0.5,1)
0.5
1
AlgEquiv
significantfigures(1.5,1)
2.0
1
AlgEquiv
significantfigures(2.5,1)
3.0
1
AlgEquiv
significantfigures(3.5,1)
4.0
1
AlgEquiv
significantfigures(11.5,2)
12.0
1
AlgEquiv
1500
scientific_notation(1500,3)
1
AlgEquiv
1500
displaysci(1.5,2,3)
1
AlgEquiv
[3,3.1,3.14,3.142,3.1416,3.141
59,3.141593,3.1415927]
makelist(significantfigures(%p
i,i),i,8)
1
Sets
AlgEquiv
x
{1,2,3}
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
co(1,2)
{1,2,3}
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
{1,2}
{1,2,3}
0 Your set should have $$3$$ different elements, but it actually has $$2$$. ATSet_wrongsz.
AlgEquiv
{2/4, 1/3}
{1/2, 1/3}
1
AlgEquiv
{A[1],A[2],A[4]}
{A[1],A[2],A[3]}
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
{A[1],A[2],A[3]}
{A[1],A[2],A[3]}
1
AlgEquiv
{1,2,4}
{1,2,3}
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
{1,x>4}
{4<x, 1}
1
AlgEquiv
{x-1=0,x>1 and 5>x}
{x>1 and x<5,x=1}
1
AlgEquiv
{x-1=0,x>1 and 5>x}
{x>1 and x<5,x=2}
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
{x-1=0,x>1 and 5>x}
{x>1 and x<3,x=1}
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 !
{-sqrt(2)/sqrt(3)}
{-2/sqrt(6)}
-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 !
{[-sqrt(2)/sqrt(3),0],[2/sqrt(
6),0]}
{[2/sqrt(6),0],[-2/sqrt(6),0]}
-3 The following entries are incorrect, although they may appear in a simplified form from that which you actually entered. $\left \{\left[ -\frac{\sqrt{2}}{\sqrt{3}} , 0 \right] \right \}$ ATSet_wrongentries.
AlgEquiv
ev(radcan({-sqrt(2)/sqrt(3)}),
simp)
ev(radcan({-2/sqrt(6)}),simp)
1
AlgEquiv
ev(radcan(ratsimp({(-sqrt(10)/
2)-2,sqrt(10)/2-2},algebraic:t
rue)),simp)
ev(radcan(ratsimp({(-sqrt(5)/s
qrt(2))-2,sqrt(5)/sqrt(2)-2},a
lgebraic:true)),simp)
1
AlgEquiv
{(2-2^(5/2))/2,(2^(5/2)+2)/2}
{1-2^(3/2),2^(3/2)+1}
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
ev(radcan({(2-2^(5/2))/2,(2^(5
/2)+2)/2}),simp)
{1-2^(3/2),2^(3/2)+1}
1
AlgEquiv
{(x-a)^6000}
{(a-x)^6000}
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
{(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
Matrices
AlgEquiv
matrix([1,2],[2,3])
matrix([1,2],[2,3])
1
AlgEquiv
matrix([1,2],[2,3])
matrix([1,2,3],[2,3,3])
0 Your matrix should be $$2$$ by $$3$$, but it is actually $$2$$ by $$2$$. ATMatrix_wrongsz_columns.
AlgEquiv
matrix([1,2],[2,3])
matrix([1,2],[2,5])
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
matrix([0.33,1],[1,1])
matrix([0.333,1],[1,1])
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
matrix([x+x,2],[2,x*x])
matrix([2*x,2],[2,x^2])
1
AlgEquiv
matrix([epsilon[0],2],[2,x^2])
matrix([epsilon[0],2],[2,x^2])
1
AlgEquiv
matrix([epsilon[2],2],[2,x^2])
matrix([epsilon[0],2],[2,x^3])
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
matrix([x>4,{1,x^2}],[[1,2]
,[1,3]])
matrix([4-x<0,{x^2, 1}],[[1
,2],[1,3]])
1
AlgEquiv
matrix([x>4,{1,x^2}],[[1,2]
,[1,3]])
matrix([4-x<0,{x^2, 1}],[[1
,2],[1,4]])
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
a
stackvector(a)
0
Equations
AlgEquiv
1
x=1
AlgEquiv
x=1
x=1
1 ATEquation_sides
AlgEquiv
1=x
1=x
1 ATEquation_sides
AlgEquiv
1=x
x=1
1 ATEquation_sides_op
AlgEquiv
1=1
1=x
0 ATEquation_default
AlgEquiv
1=1
x=1
0 ATEquation_default
AlgEquiv
x=2
x=1
0 ATEquation_lhs_notrhs
AlgEquiv
2=x
x=1
0 ATEquation_default
AlgEquiv
x=x
y=y
1 ATEquation_zero
AlgEquiv
x+y=1
y=1-x
1
AlgEquiv
2*x+2*y=1
y=0.5-x
1 ATEquation_ratio
AlgEquiv
1/x+1/y=2
y = x/(2*x-1)
1 ATEquation_ratio
AlgEquiv
y=sin(2*x)
y/2=cos(x)*sin(x)
1 ATEquation_ratio
AlgEquiv
y=(x-a)^6000
y=(x-a)^6000
1 ATEquation_sides
AlgEquiv
y=(x-a)^5999
y=(x-a)^6000
0 ATEquation_lhs_notrhs
AlgEquiv
y=(a-x)^6000
y=(x-a)^6000
1 ATEquation_sides
AlgEquiv
y=(a-x)^5999
y=(x-a)^5999
0 ATEquation_lhs_notrhs
AlgEquiv
y=(a-x)^59999
y=(x-a)^5999
0 ATEquation_lhs_notrhs
AlgEquiv
x+y=i
y=i-x
1
AlgEquiv
(1+%i)*(x+y)=0
y=-x
1
AlgEquiv
s^2*%e^(s*t)=0
s^2=0
0 ATEquation_default
AlgEquiv
0=-x+y/A+(y-z)/B
0=x-y/A-(y-z)/B
1
AlgEquiv
x^6000-x^6001=x^5999
x^5999*(1-x+x^2)=0
1 ATEquation_ratio
AlgEquiv
x^6000-x^6001=x^5999
x^5999*(1-x+x^3)=0
0 ATEquation_default
AlgEquiv
258552*x^7*(81*x^8+1)^398
x^3*(x^4+1)^399
0
AlgEquiv
Ia*(R1+R2+R3)-Ib*R3=0
-Ia*(R1+R2+R3)+Ib*R3=0
1
AlgEquiv
a=0 or b=0
a*b=0
1 ATEquation_sides
AlgEquiv
a*b=0
a=0 or b=0
1 ATEquation_sides
AlgEquiv
a*x=a*y
x=y
0 ATEquation_default
AlgEquiv
a*x=a*y
a=0 or x=y
1 ATEquation_ratio
Unary Equations
AlgEquiv
1
stackeq(1)
1
AlgEquiv
stackeq(1)
1
1
AlgEquiv
stackeq(1)
0
0
Equations: Loose/gain roots with nth powers of each side.
AlgEquiv
x=y
x^2=y^2
0 ATEquation_default
AlgEquiv
(x-2)^2=0
x=2
0 ATEquation_default
AlgEquiv
4*x^2-71*x+220 = 0 or 14*x^2-9
1*x+140 = 0
x = 5/2 or x = 4 or x = 55/4
0 ATEquation_default
AlgEquiv
4*x^2-71*x+220 = 0 or 14*x^2-9
1*x+140 = 0
x = 5/2 or x = 4 or x=4 or x =
55/4
1 ATEquation_sides
AlgEquiv
x^2=4
x=2 or x=-2
1 ATEquation_ratio
AlgEquiv
a^3*b^3=0
a=0 or b=0
0 ATEquation_default
AlgEquiv
a^3*b^3=0
a*b=0
0 ATEquation_default
AlgEquiv
(x-y)*(x+y)=0
x^2=y^2
1 ATEquation_ratio
AlgEquiv
x=1
(x-1)^3=0
0 ATEquation_default
AlgEquiv
sqrt(x)=sqrt(y)
x=y
0 ATEquation_default
AlgEquiv
x=sqrt(a)
x^2=a
0 ATEquation_default
AlgEquiv
(x-sqrt(a))*(x+sqrt(a))=0
x^2=a
1 ATEquation_ratio
AlgEquiv
(x-%i*sqrt(a))*(x+%i*sqrt(a))=
0
x^2=-a
1 ATEquation_ratio
AlgEquiv
(x-%i*sqrt(abs(a)))*(x+%i*sqrt
(abs(a)))=0
x^2=-abs(a)
1 ATEquation_ratio
AlgEquiv
y=sqrt(1-x^2)
x^2+y^2=1
0 ATEquation_default
AlgEquiv
(y-sqrt(1-x^2))*(y+sqrt(1-x^2)
)=0
x^2+y^2=1
1 ATEquation_ratio
AlgEquiv
(y-sqrt((1-x)*(1+x)))*(y+sqrt(
(1-x)*(1+x)))=0
x^2+y^2=1
1 ATEquation_ratio
AlgEquiv
(x-1)*(x+1)*(y-1)*(y+1)=0
y^2+x^2=1+x^2*y^2
1 ATEquation_ratio
Equations: edge cases. Teacher must enter an equation, all or none here.
AlgEquiv
all
x=x
1 ATEquation_zero
AlgEquiv
true
x=x
1 ATEquation_zero
AlgEquiv
x=x
all
1 ATEquation_zero
AlgEquiv
all
all
1 ATEquation_zero
AlgEquiv
true
all
1 ATEquation_zero
AlgEquiv
a=a
x=x
1 ATEquation_zero
AlgEquiv
false
x=x
0 ATEquation_zero_fail
AlgEquiv
false
all
0 ATEquation_zero_fail
AlgEquiv
none
all
0 ATEquation_zero_fail
AlgEquiv
all
none
0 ATEquation_empty_fail
AlgEquiv
2=3
1=4
1 ATEquation_empty
AlgEquiv
none
1=2
1 ATEquation_empty
AlgEquiv
false
1=2
1 ATEquation_empty
AlgEquiv
none
none
1 ATEquation_empty
AlgEquiv
false
none
1 ATEquation_empty
AlgEquiv
3=0
none
1 ATEquation_empty
AlgEquiv
0=3
none
1 ATEquation_empty
AlgEquiv
all
1=2
0 ATEquation_empty_fail
AlgEquiv
true
1=2
0 ATEquation_empty_fail
AlgEquiv
{}
1=2
AlgEquiv
[]
1=2
AlgEquiv
{}
none
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
x^2
cc(1,3)
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
%union(oo(1,2),oo(3,4))
%union(oo(1,2),oo(3,4))
1 ATRealSet_true.
AlgEquiv
%union(oc(1,2),co(2,3))
oo(1,3)
1 ATRealSet_true.
AlgEquiv
%union(oc(1,2),co(2,3))
cc(1,3)
0 ATRealSet_false.
AlgEquiv
{-1,1}
%union({-1,1})
1 ATRealSet_true.
AlgEquiv
{1,3}
cc(1,3)
0 ATRealSet_false.
AlgEquiv
%intersection(oc(-1,1),co(1,2)
)
%union({1})
1 ATRealSet_true.
AlgEquiv
oo(-inf,1)
oo(-inf,1)
1 ATRealSet_true.
AlgEquiv
oo(-1,inf)
oo(0,inf)
0 ATRealSet_false.
AlgEquiv
%union(oc(-inf,0),oo(-1,4))
oo(-inf,4)
1 ATRealSet_true.
AlgEquiv
%union(oo(-inf,1),oo(-1,inf))
oo(-inf,inf)
1 ATRealSet_true.
AlgEquiv
all
oo(-inf,inf)
1 ATRealSet_true.
AlgEquiv
co(1,2)
1 <= x nounand x<2
0 Your answer should be an equation, inequality or a logical combination of many of these, but is not. ATAlgEquiv_SA_not_logic.
AlgEquiv
1 <= x nounand x<2
co(1,2)
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
minf <= x
co(minf,inf)
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
-inf <= x
co(minf,inf)
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
x <= inf
oc(minf,inf)
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
minf <= x
oo(minf,inf)
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
single_variable_solver_real(mi
nf <= x)
co(minf,inf)
1 ATRealSet_true.
AlgEquiv
single_variable_solver_real(-i
nf <= x)
co(minf,inf)
1 ATRealSet_true.
AlgEquiv
single_variable_solver_real(x
<= inf)
oc(minf,inf)
1 ATRealSet_true.
AlgEquiv
single_variable_solver_real(mi
nf <= x)
oo(minf,inf)
0 ATRealSet_false.
Complex numbers
AlgEquiv
a=b/%i
%i*a=b
1 ATEquation_num_i
AlgEquiv
b/%i=a
%i*a=b
1 ATEquation_num_i
AlgEquiv
b=a/%i
%i*a=b
0 ATEquation_lhs_notrhs_op
AlgEquiv
a*(2+%i)=b
a=b/(2+%i)
1 ATEquation_ratio
AlgEquiv
a*(2+%i)=b
a=b*(2-%i)/5
1 ATEquation_num_i
AlgEquiv
a*(2+%i)=b
a=b*(2-%i)/4
0 ATEquation_default
AlgEquiv
i
disp_complex(0,1)
0
Absolute value in equations
AlgEquiv
abs(x)=abs(y)
x=y
0 ATEquation_default
AlgEquiv
abs(x)=abs(y)
x=y or x=-y
1
AlgEquiv
abs(x)=abs(y)
(x-y)*(x+y)=0
1
Functions
AlgEquiv
f(x):=1/0
f(x):=x^2
AlgEquiv
1
f(x):=x^2
0 Your answer should be a function, defined using the operator :=, but is not. ATAlgEquiv_SA_not_function.
AlgEquiv
f(x)=x^2
f(x):=x^2
0 Your answer should be a function, defined using the operator :=, but is not. ATAlgEquiv_SA_not_function.
AlgEquiv
f(x):=x^2
f(x,y):=x^2+y^2
0 ATFunction_length_args. ATFunction_false.
AlgEquiv
f(x):=x^2
f(x)=x^2
AlgEquiv
f(x):=x^2
f(x):=x^2
1 ATFunction_true.
AlgEquiv
f(x):=x^2
f(x):=sin(x)
0 ATFunction_false.
AlgEquiv
g(x):=x^2
f(x):=x^2
0 ATFunction_wrongname. ATFunction_true.
AlgEquiv
f(y):=y^2
f(x):=x^2
1 ATFunction_arguments_different. ATFunction_true.
AlgEquiv
f(a,b):=a^2+b^2
f(x,y):=x^2+y^2
1 ATFunction_arguments_different. ATFunction_true.
Inequalities
AlgEquiv
1
x>1
AlgEquiv
x=1
x>1 and x<5
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
x<1
x>1
0 Your inequality appears to be backwards. ATInequality_backwards.
AlgEquiv
1<x
x>1
1
AlgEquiv
a<b
b>a
1
AlgEquiv
2<2*x
x>1
1
AlgEquiv
-2>-2*x
x>1
1
AlgEquiv
x>1
x<=1
0 Your inequality should not be strict! Your inequality appears to be backwards. ATInequality_strict. ATInequality_backwards.
AlgEquiv
x>=2
x<2
0 Your inequality should be strict, but is not! Your inequality appears to be backwards. ATInequality_nonstrict. ATInequality_backwards.
AlgEquiv
x>=1
x>2
0 Your inequality should be strict, but is not! ATInequality_nonstrict.
AlgEquiv
x>1
x>1
1
AlgEquiv
x>=1
x>=1
1
AlgEquiv
x>2
x>1
0
AlgEquiv
1<x
x>1
1
AlgEquiv
2*x>=x^2
x^2<=2*x
1
AlgEquiv
2*x>=x^2
x^2<=2*x
1
AlgEquiv
3*x^2<9*a
x^2-3*a<0
1
AlgEquiv
x^2>4
x>2 or x<-2
1 ATLogic_True.
AlgEquiv
1<x or x<-3
x<-3 or 1<x
1 ATLogic_True.
AlgEquiv
1<x or x<-3
x<-1 or 3<x
0
AlgEquiv
x>1 and x<5
x>1 and x<5
1 ATLogic_True.
AlgEquiv
x>1 and x<5
5>x and 1<x
1 ATLogic_True.
AlgEquiv
not (x<=2 and -2<=x)
x>2 or -2>x
1 ATLogic_True.
AlgEquiv
x>2 or -2>x
not (x<=2 and -2<=x)
1 ATLogic_True.
AlgEquiv
x>=1 or 1<=x
x>=1
1
AlgEquiv
x>=1 and x<=1
x=1
1 ATInequality_solver.
AlgEquiv
(x>4 and x<5) or (x<-
4 and x>-5) or (x+5>0 an
d x<-4)
(x>-5 and x<-4) or (x>
;4 and x<5)
1 ATLogic_True.
AlgEquiv
(x>4 and x<5) or (x<-
4 and x>-5) or (x+5>0 an
d x<-4)
(x>-5 and x<-4) or (x>
;8 and x<5)
0
AlgEquiv
(x < 0 nounor x >= 1) no
unand x <= 3
x < 0 or (x >= 1 and x &
lt;= 3)
1 ATLogic_True.
AlgEquiv
(x < 0 nounor x >= 1) no
unand x <= 3
x < 0 or x >= 1 and x &l
t;= 3
1 ATLogic_True.
AlgEquiv
(x < 0 nounor x >= 1) no
unand x <= 3
x < 0 or (x >= 1 and x &
lt;= 3)
1 ATLogic_True.
AlgEquiv
(x < 0 nounor x >= 1) no
unand x <= 3
(x < 0 or x >= 1) and x
<= 3
1 ATLogic_True.
AlgEquiv
(x < 0 nounor x >= 1) no
unand x <= 3
x < 0 or (x >= 1 and x &
lt;= 3)
1 ATLogic_True.
AlgEquiv
natural_domain(1/x^2)
natural_domain(1/x)
1 ATRealSet_true.
AlgEquiv
x^4>=0
x^2>=0
1
AlgEquiv
x^4>=16
x^2>=4
1
AlgEquiv
x^4>=16
x^2>=4
1
AlgEquiv
-3<=x
-3<=x nounand x<=3
0
AlgEquiv
{2,-2}
x>2 nounor -2>x
0 Your answer should be an equation, inequality or a logical combination of many of these, but is not. ATAlgEquiv_SA_not_logic.
AlgEquiv
x^2<4
x<2 nounand x>-2
1 ATLogic_Solver_True.
AlgEquiv
x^2<6
x<2 nounand x>-2
0
AlgEquiv
x>1 nounand x<-1
false
1 ATLogic_Solver_True.
AlgEquiv
x>1 nounand x<3
true
0
AlgEquiv
x>1 nounor x<3
true
1 ATLogic_Solver_True.
AlgEquiv
x>1 nounor x<3
all
1 ATLogic_Solver_True.
AlgEquiv
abs(x)<1
abs(x)<1
1
AlgEquiv
abs(x)<1
abs(x)<2
0
AlgEquiv
abs(x)<1
abs(x)>1
0 Your inequality appears to be backwards. ATInequality_backwards.
AlgEquiv !
abs(x)<2
-2<x and x<2
-3
AlgEquiv !
-2<x and x<2
abs(x)<2
-3
AlgEquiv
abs(x)<2
-1<x and x<1
0
AlgEquiv
x^2<=9
abs(x)<3
0
AlgEquiv !
x^2<=9
abs(x)<=3
-3
AlgEquiv !
x^6<1
abs(x)<1
-3
AlgEquiv !
abs(x)>1
x<-1 or x>1
-3
AlgEquiv
minf < x
minf <= x
0 Your inequality should not be strict! ATInequality_strict.
AlgEquiv
x>minf
minf < x
1
AlgEquiv
x>-inf
minf < x
1
AlgEquiv
x<2*inf
x<inf
0
AlgEquiv
minf < x nounand x <1
x<1
1
AlgEquiv
minf < x nounand x <1
x<2
0
Maxima and infinity
AlgEquiv
2*inf
inf
0
AlgEquiv
-inf
minf
0
Not equal to
AlgEquiv
x#1
x#1
1 ATLogic_True.
AlgEquiv
x#(1+1)
x#2
1 ATLogic_True.
AlgEquiv
1#x
x#1
1 ATLogic_True.
AlgEquiv
x#2
x-2#0
1 ATLogic_True.
AlgEquiv
[x#2]
[x-2#0]
1
AlgEquiv
x-3#0
x#2
0
AlgEquiv
x#2
x<2 nounor x>2
1 ATLogic_Solver_True.
AlgEquiv
x^2-3#1
x<-2 nounor (x<-2 and x&
lt;2) nounor 2<x
0
AlgEquiv
x^2-3#1
x<-2 nounor (-2<x and x&
lt;2) nounor 2<x
1 ATLogic_Solver_True.
AlgEquiv
x#1
x#0
0
Surds
AlgEquiv
sqrt(12)
2*sqrt(3)
1
AlgEquiv
sqrt(11+6*sqrt(2))
3+sqrt(2)
1
AlgEquiv
(19601-13860*sqrt(2))^(7/4)
(5*sqrt(2)-7)^7
1
AlgEquiv
(19601-13861*sqrt(2))^(7/4)
(5*sqrt(2)-7)^7
0
AlgEquiv
(19601-13861*sqrt(2))^(7/4)
(5*sqrt(2)-7)^7
0
AlgEquiv
sqrt(2*log(26)+4-2*log(2))
sqrt(2*log(13)+4)
1
AlgEquiv
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
Factorials and binomials
AlgEquiv
(n+1)*n!
(n+1)!
1
AlgEquiv
n/n!
1/(n-1)!
1
AlgEquiv
n/n!
1/(n+1)!
0
AlgEquiv
n!/(k!*(n-k)!)
binomial(n,k)
1
AlgEquiv !
binomial(n,k)+binomial(n,k+1)
binomial(n+1,k+1)
-3
AlgEquiv
binomial(n,k)+binomial(n,k+1)
binomial(n+1,k)
0
AlgEquiv
binomial(n,k)
binomial(n,n-k)
1
AlgEquiv
175!*56!/(55!*176!)
17556/55176
1
Unevaluated derviatives
AlgEquiv
3*s*diff(q(s),s)
3*s*diff(q(s),s)
1
Sums and products
AlgEquiv
sum(k^n,n,0,3)
sum(k^n,n,0,3)
1
AlgEquiv
1+k+k^2+k^3
sum(k^n,n,0,3)
1
AlgEquiv
1+k+k^2
sum(k^n,n,0,3)
0
AlgEquiv
n*(n+1)*(2*n+1)/6
sum(k^2,k,1,n)
1
AlgEquiv
sum((k+1)^2,k,0,n-1)
sum(k^2,k,1,n)
1
AlgEquiv
product(cos(k*x),k,1,3)
product(cos(k*x),k,1,3)
1
AlgEquiv
cos(x)*cos(2*x)*cos(3*x)
product(cos(k*x),k,1,3)
1
AlgEquiv
cos(x)*cos(2*x)
product(cos(k*x),k,1,3)
0
Scientific units are ignored
AlgEquiv
9.81*m/s^2
stackunits(9.81,m/s^2)
1
AlgEquiv
6*stackunits(1,m)
stackunits(6,m)
1
AlgEquiv
stackunits(2,m)^2
stackunits(4,m^2)
1
AlgEquiv
stackunits(2,s)^2
stackunits(4,m^2)
0
Maxima does not simplify -inf (I agree!)
AlgEquiv
-inf
minf
0
These currently fail
AlgEquiv !
2/%i*ln(sqrt((1+z)/2)+%i*sqrt(
(1-z)/2))
-%i*ln(z+%i*sqrt(1-z^2))
-3
AlgEquiv !
abs(x^2-4)/(abs(x-2)*abs(x+2))
1
-3
AlgEquiv !
abs(x^2-4)
abs(x-2)*abs(x+2)
-3
AlgEquiv !
(-1)^n*cos(x)^n
(-cos(x))^n
-3
AlgEquiv !
(sqrt(108)+10)^(1/3)-(sqrt(108
)-10)^(1/3)
2
-3
AlgEquiv !
(sqrt(2+sqrt(2))+sqrt(2-sqrt(2
)))/(2*sqrt(2))
sqrt(sqrt(2)+2)/2
-3
AlgEquiv !
sqrt(2*x*sqrt(x^2+1)+2*x^2+1)-
sqrt(x^2+1)-x
0
-3
AlgEquiv !
(77+20*sqrt(13))^(1/6)-(77-20*
sqrt(13))^(1/6)
1
-3
AlgEquiv !
(930249+416020*sqrt(5))^(1/30)
-(930249-416020*sqrt(5))^(1/30
)
1
-3
AlgEquiv !
cos(2*%pi/17)
(-1+sqrt(17)+sqrt(34-2*sqrt(17
)))/16+(2*sqrt(17+3*sqrt(17)-s
qrt(34-2*sqrt(17))-2*sqrt(34+2
*sqrt(17))))/16
-3
AlgEquiv !
(41-sqrt(511))/2
(sqrt((4*(cos((1/2*(acos((61/1
040*sqrt(130)))-atan(11/3)))))
^(2))+21)-(2*cos((1/2*(acos((6
1/1040*sqrt(130)))-atan(11 / 3
))))))^(2)
-3
AlgEquiv !
a*(1+sqrt(2))=b
a=b*(sqrt(2)-1)/3
-3 ATEquation_default
This is only equivalent for x>=0...
AlgEquiv !
atan(1/2)
%pi/2-atan(2)
-3
This is true for all x...
AlgEquiv !
asinh(x)
ln(x+sqrt(x^2+1))
-3
Logical expressions
AlgEquiv
true and false
false
1 ATLogic_True.
AlgEquiv
true or false
false
0
AlgEquiv
A and B
B and A
1 ATLogic_True.
AlgEquiv
A and B
C and A
0
AlgEquiv
A and B=C
C=B and A
1 ATLogic_True.
AlgEquiv
A and (B and C)
A and B and C
1 ATLogic_True.
AlgEquiv
A and (B or C)
A and (B or C)
1 ATLogic_True.
AlgEquiv
(A and B) or (A and C)
A and (B or C)
1 ATLogic_True.
AlgEquiv
-(b#pm#sqrt(b^2-4*a*c))
-b#pm#sqrt(b^2-4*a*c)
1 ATLogic_True.
AlgEquiv
x=-b#pm#c^2
x=c^2-b or x=-c^2-b
1 ATLogic_True.
AlgEquiv
x#pm#a = y#pm#b
x#pm#a = y#pm#b
1 ATEquation_sides
AlgEquiv
x#pm#a = y#pm#b
x#pm#a = y#pm#c
0 ATEquation_lhs_notrhs
AlgEquiv
not(A) and not(B)
not(A or B)
1 ATLogic_True.
AlgEquiv
not(A) and not(B)
not(A and B)
0
AlgEquiv
not(A) or B
boolean_form(A implies B)
1
AlgEquiv
not(A) or B
A implies B
1 ATLogic_True.
AlgEquiv
not(A) and B
A implies B
0
AlgEquiv
(not A and B) or (not B and A)
A xor B
1 ATLogic_True.
AlgEquiv
(A and B) or (not A and not B)
A xnor B
1 ATLogic_True.
AlgEquiv
{not(A) or B,A and B}
{A implies B,A and B}
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
{A implies B,A and B}
{not(A) and B,A and B}
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
diff(x^2,x)
2*x
1
AlgEquiv
diff(x^2,x)
'diff(x^2,x)
1
AlgEquiv
noundiff(x^2,x)
2*x
1
AlgEquiv
diff(y,x)
0
1
AlgEquiv
noundiff(y,x)
0
1
AlgEquiv
diff(y(x),x)
0
0
AlgEquiv
diff(y(x),x)
diff(y,x)
0
AlgEquiv
diff(y,x)
diff(y,x,2)
1
Basic support for strings
AlgEquiv
"Hello"
"Hello"
1 ATAlgEquiv_String
AlgEquiv
"hello"
"Hello"
0 ATAlgEquiv_String
AlgEquiv
W
"Hello"
AlgEquiv
"Hello"
x^2
0 Your answer should be an expression, not an equation, inequality, list, set or matrix. ATAlgEquiv_SA_not_expression.

## AlgEquivNouns

Test
?
Student response
Opt
Mark
CAS errors
Feedback
AlgEquivNouns
1/0
1
AlgEquivNouns
1
1/0
AlgEquivNouns
(x-1)^2
AlgEquivNouns
x^2
AlgEquivNouns
x-1)^2
(x-1)^2
AlgEquivNouns
diff(x^2,x)
2*x
1
AlgEquivNouns
diff(x^2,x)
'diff(x^2,x)
0
AlgEquivNouns
diff(x^2,x)
'diff(x^2,x)
0
AlgEquivNouns
'diff(y,x)
noundiff(y,x)
1
AlgEquivNouns
diff(y,x)
0
1
AlgEquivNouns
'diff(y,x)
0
0
AlgEquivNouns
noundiff(y,x)
0
0
AlgEquivNouns
diff(y(x),x)
0
0
AlgEquivNouns
'diff(y,x,1)
'diff(y,x,2)
0
AlgEquivNouns
'diff(y(x),x)
'diff(y,x)
0
AlgEquivNouns
subst(y,y(x),'diff(y,x)+y
=1)
'diff(y,x)+y=1
1 ATEquation_sides
AlgEquivNouns
subst(y,y(x),'diff(y(x),x
)+y(x)=1)
'diff(y,x)+y=1
1 ATEquation_sides
AlgEquivNouns
subst(y(x),y,'diff(y,x)+y
=1)
'diff(y(x),x)+y(x)=1
1 ATEquation_sides
AlgEquivNouns
subst(y(x),y,'diff(y,x)+y
=1)
'diff(y,x)+y=1
0 ATEquation_default
AlgEquivNouns
subst(y(x),y,'diff(y(x),x
)+y(x)=1)
'diff(y,x)+y=1
-1 TEST_FAILED The answer test failed to execute correctly: please alert your teacher. subst: cannot substitute y(x) for operator y in expression y(x) ATAlgEquivNouns_STACKERROR_SAns.
AlgEquivNouns
y_x
'diff(y,x)
0
Partials
AlgEquivNouns
noundiff(f,x,1,y,1)
noundiff(noundiff(f,x),y)
1
AlgEquivNouns
noundiff(noundiff(f,y),x)
noundiff(noundiff(f,x),y)
1
AlgEquivNouns
noundiff(noundiff(f,x),x)
noundiff(f,x,2)
1
Differential equations
AlgEquivNouns
noundiff(H,x,2) = -R/T
noundiff(H,x,2) + R/T = 0
1 ATEquation_ratio
AlgEquivNouns
'diff(H,x,2) = -R/T
noundiff(H,x,2) + R/T = 0
1 ATEquation_ratio
AlgEquivNouns
y(t)=int(s^2,s,0,t)
y(t)=t^3/3
1 ATEquation_sides
AlgEquivNouns
y(t)='int(s^2,s,0,t)
y(t)=t^3/3
0 ATEquation_lhs_notrhs
AlgEquivNouns
y(t)='int(s^2,s,0,t)
y(t)=nounint(s^2,s,0,t)
1 ATEquation_sides
Logic nouns are still evaluated
AlgEquivNouns
true nounand false
false
1 ATLogic_True.

## SubstEquiv

Test
?
Student response
Opt
Mark
CAS errors
Feedback
SubstEquiv
1/0
x^2-2*x+1
SubstEquiv
x^2
x^2-2*x+1
[1/0]
SubstEquiv
x^2
x^2-2*x+1
x
SubstEquiv
x^2+1
x^2+1
1
SubstEquiv
x^2+1
x^3+1
0
SubstEquiv
x^2+1
x^3+1
0
SubstEquiv
X^2+1
x^2+1
1 Your answer would be correct if you used the following substitution of variables. $\left[ X=x \right]$ ATSubstEquiv_Subst [X = x].
SubstEquiv
x^2+y
a^2+b
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
x^2+y/z
a^2+c/b
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
y=x^2
a^2=b
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
{x=1,y=2}
{x=2,y=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
cos(a*x)/(x*(ln(x)))
cos(a*y)/(y*(ln(y)))
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
cos(a*x)/(x*(ln(x)))
cos(x*a)/(a*(ln(a)))
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
cos(a*x)/(x*(ln(x)))
cos(a*x)/(x(ln(x)))
0
SubstEquiv
cos(a*x)/(x*(ln(x)))
cos(a*y)/(y(ln(y)))
0
SubstEquiv
x+1>y
y+1>x
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
x+1>y
x<y+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
matrix([1,A^2+A+1],[2,0])
matrix([1,x^2+x+1],[2,0])
1 Your answer would be correct if you used the following substitution of variables. $\left[ A=x \right]$ ATSubstEquiv_Subst [A = x].
SubstEquiv
matrix([B,A^2+A+1],[2,C])
matrix([y,x^2+x+1],[2,z])
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
matrix([B,A^2+A+1],[2,C])
matrix([y,x^2+x+1],[2,x])
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
[x^2+1,x^2]
[A^2+1,A^2]
1 Your answer would be correct if you used the following substitution of variables. $\left[ x=A \right]$ ATSubstEquiv_Subst [x = A].
SubstEquiv
[x^2-1,x^2]
[A^2+1,A^2]
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
[A,B,C]
[B,C,A]
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
[A,B,C]
[B,B,A]
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
[1,[A,B],C]
[1,[a,b],C]
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
{x^2+1,x^2}
{A^2+1,A^2}
1 Your answer would be correct if you used the following substitution of variables. $\left[ x=A \right]$ ATSubstEquiv_Subst [x = A].
SubstEquiv
{x^2-1,x^2}
{A^2+1,A^2}
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
{A+1,B^2,C}
{B,C+1,A^2}
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
{1,{A,B},C}
{1,{a,b},C}
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
A*cos(t)+B*sin(t)
P*cos(t)+Q*sin(t)
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
A*cos(t)+B*sin(t)
P*cos(x)+Q*sin(x)
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
A*cos(t)+B*sin(t)
P*cos(x)+Q*sin(x)
[x]
0
SubstEquiv
A*cos(t)+B*sin(t)
P*cos(x)+Q*sin(x)
[t]
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
A*cos(t)*e^x+B*sin(t)*e^x+C*si
n(2*x)+D*cos(2*x)
P*cos(t)*e^x+Q*sin(t)*e^x+R*si
n(2*x)+S*cos(2*x)
[x,t]
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].
SubstEquiv
sqrt(2*g*y)
sqrt(2*g*x)
1 Your answer would be correct if you used the following substitution of variables. $\left[ g=g , y=x \right]$ ATSubstEquiv_Subst [g = g,y = x].
SubstEquiv
sqrt(2*g*y)
sqrt(2*g*x)
[g]
1 Your answer would be correct if you used the following substitution of variables. $\left[ y=x \right]$ ATSubstEquiv_Subst [y = x].

## EqualComAss

Test
?
Student response
Opt
Mark
CAS errors
Feedback
EqualComAss
1/0
0
-1 ATEqualComAss_STACKERROR_SAns.
EqualComAss
0
1/0
-1 ATEqualComAss_STACKERROR_TAns.
Numbers
EqualComAss
2/4
1/2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
3^2
8
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
3^2
9
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
cos(0)
1
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
4^(1/2)
2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
1/3^(1/2)
(1/3)^(1/2)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
sqrt(3)/3
(1/3)^(1/2)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
sqrt(3)
3^(1/2)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
2*sqrt(2)
sqrt(8)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
2*2^(1/2)
sqrt(8)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
sqrt(2)/4
1/sqrt(8)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
1/sqrt(2)
2^(1/2)/2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
4.0
4
0 ATEqualComAss (AlgEquiv-true).
Case sensitivity
EqualComAss
X
x
0 ATEqualComAss (AlgEquiv-false)ATAlgEquiv_WrongCase.
EqualComAss
exdowncase(X)
x
1
EqualComAss
exdowncase((X-1)^2)
x^2-2*x+1
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
exdowncase(X^2-2*X+1)
x^2-2*x+1
1
Powers
EqualComAss
a^2/b^3
a^2*b^(-3)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
lg(a^x,a)
x
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
x^(2/4)
x^(1/2)
0 ATEqualComAss (AlgEquiv-true).
Simple polynomials
EqualComAss
1+2*x
x*2+1
1
EqualComAss
1+x
2*x+1
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
1+x+x
2*x+1
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(x+y)+z
z+x+y
1
EqualComAss
x*x
x^2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(x+5)*x
x*(5+x)
1
EqualComAss
x*(x+5)
5*x+x^2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(1-x)^2
(x-1)^2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(a-x)^6000
(x-a)^6000
0 ATEqualComAss (AlgEquiv-true).
Expressions with subscripts
EqualComAss
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
EqualComAss
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 ATEqualComAss (AlgEquiv-false).
Unary minus
EqualComAss
-1+2
2-1
1
EqualComAss
-1*2+3*4
3*4-1*2
1
EqualComAss
(-1*2)+3*4
10
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
-1*2+3*4
3*4-1*2
1
EqualComAss
x*(-y)
-x*y
1
EqualComAss
x*(-y)
-(x*y)
1
EqualComAss
(-x)*(-x)
x*x
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(-x)*(-x)
x^2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
-1/4*%pi*i
-(%i*%pi)/4
0 ATEqualComAss (AlgEquiv-true).
Rational expressions
EqualComAss
1/2
3/6
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
1/(1+2*x)
1/(2*x+1)
1
EqualComAss
2/(4+2*x)
1/(x+2)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(a*b)/c
a*(b/c)
1
EqualComAss
((x+1)/(x*(x-1)))*(x-1)
((x+1)*(x-1))/(x*(x-1))
1
EqualComAss
(-x)/y
-(x/y)
1
EqualComAss
x/(-y)
-(x/y)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
-1/(1-x)
1/(x-1)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
1/2*1/x
1/(2*x)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(k+8)/(k^2+4*k-12)
(k+8)/(k^2+4*k-12)
1
EqualComAss
(k+8)/(k^2+4*k-12)
(k+8)/((k-2)*(k+6))
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(k+7)/(k^2+4*k-12)
(k+8)/(k^2+4*k-12)
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
-(2*k+6)/(k^2+4*k-12)
-(2*k+6)/(k^2+4*k-12)
1
EqualComAss
(a+b)/1
(b+a)/1
1
No simplicifcation here
EqualComAss
1*x
x
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
23+0*x
23
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
x+0
x
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
x^1
x
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(1/2)*(a+b)
(a+b)/2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
1/3*logbase(27,6)
logbase(27,6)/3
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
1/3*lg(27,6)
lg(27,6)/3
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
lg(root(x, n))
lg(x, 10)/n
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
exp(x)
%e^x
1
EqualComAss
exp(x)^2
%e^(2*x)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
exp(x)^2
(%e^(x))^2
1
EqualComAss
1/3*i
i/3
0 ATEqualComAss (AlgEquiv-true).
Complex numbers
EqualComAss
%i
e^(i*pi/2)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(4*sqrt(3)*%i+4)^(1/5)
rectform((4*sqrt(3)*%i+4)^(1/5
))
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(4*sqrt(3)*%i+4)^(1/5)
8^(1/5)*(cos(%pi/15)+%i*sin(%p
i/15))
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
(4*sqrt(3)*%i+4)^(1/5)
polarform((4*sqrt(3)*%i+4)^(1/
5))
0 ATEqualComAss (AlgEquiv-true).
Equations
EqualComAss
y=x
x=y
1
EqualComAss
x+1
y=2*x+1
0 Your answer should be an equation, but is not. ATEqualComAss ATAlgEquiv_SA_not_equation.
EqualComAss
y=1+2*x
y=2*x+1
1
EqualComAss
y=x+x+1
y=1+2*x
0 ATEqualComAss (AlgEquiv-true).
Logic
EqualComAss
A and B
B and A
1
EqualComAss
A or B
B or A
1
EqualComAss
A or B
B and A
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
not(true)
false
0 ATEqualComAss (AlgEquiv-true).
Sets
EqualComAss
{2*x+1,2}
{2, 1+x*2}
1
EqualComAss
2
{2}
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
{2*x+1, 1+1}
{2, 1+x*2}
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
{1,2}
{1,{2}}
0 ATEqualComAss (AlgEquiv-false)ATSet_wrongentries.
EqualComAss
{4,3}
{3,4}
1
EqualComAss
{4,4}
{4}
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
{-1,1,-1}
{-1,-1,1}
1
EqualComAss
{-1,1,-1}
{-1,1}
0 ATEqualComAss (AlgEquiv-true).
Lists
EqualComAss
[2*x+1,2]
[1+x*2,2]
1
EqualComAss
[x+x+1, 1+1]
[1+x*2,2]
0 ATEqualComAss (AlgEquiv-true).
Matrices
EqualComAss
matrix([1,2],[2,3])
matrix([1,2],[2,3])
1
EqualComAss
matrix([1,2],[2,3])
matrix([1,2,3],[2,3,3])
0 ATEqualComAss (AlgEquiv-false)ATMatrix_wrongsz_columns.
EqualComAss
matrix([1,2],[2,3])
matrix([1,2],[2,5])
0 ATEqualComAss (AlgEquiv-false)ATMatrix_wrongentries.
EqualComAss
matrix([1,2],[2,2+1])
matrix([1,2],[2,3])
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
matrix([x+x, 1],[1, 1])
matrix([2*x, 1],[1, 1])
0 ATEqualComAss (AlgEquiv-true).
Sums and products
EqualComAss
sum(k^n,n,0,3)
sum(k^n,n,0,3)
1
EqualComAss
1+k+k^2+k^3
sum(k^n,n,0,3)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
sum(k,k,0,1+n)
sum(k,k,0,n+1)
1
EqualComAss
(n+1)*(n+2)/2
sum(k,k,0,n+1)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
product(cos(k*x),k,1,3)
product(cos(k*x),k,1,3)
1
EqualComAss
cos(x)*cos(2*x)*cos(3*x)
product(cos(k*x),k,1,3)
0 ATEqualComAss (AlgEquiv-true).
Inequalities are not commutative under this test
EqualComAss
-6/5 > x
x < -6/5
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
x<1 and -3<x
-3<x and x<1
1
EqualComAss
1>x and -3<x
-3<x and x<1
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
make_less_ineq(-6/5 > x)
x < -6/5
1
EqualComAss
make_less_ineq(1>x and -3&l
t;x)
-3<x and x<1
1
EqualComAss
make_less_ineq(6/3 > x)
x < 2
0 ATEqualComAss (AlgEquiv-true).
Unary Equations
EqualComAss
1
stackeq(1)
1
EqualComAss
stackeq(1)
1
1
EqualComAss
stackeq(1+1)
2
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
stackeq(1)
0
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
lowesttermsp(1/3)
true
1
EqualComAss
lowesttermsp(2/6)
true
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
lowesttermsp(x^2/x)
true
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
lowesttermsp(-y/-x)
true
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
lowesttermsp((x^2-1)/(x-1))
true
0 ATEqualComAss (AlgEquiv-false).
EqualComAss
lowesttermsp((x^2-1)/(x+2))
true
1
EqualComAss
rationalized(1+sqrt(3)/3)
true
1
EqualComAss
rationalized(1+1/sqrt(3))
[sqrt(3)]
1
EqualComAss
rationalized(1/sqrt(3))
[sqrt(3)]
1
EqualComAss
rationalized(1/sqrt(2)+i/sqrt(
2))
[sqrt(2),sqrt(2)]
1
EqualComAss
rationalized(sqrt(2)/2+1/sqrt(
3))
[sqrt(3)]
1
EqualComAss
rationalized(1/sqrt(2)+1/sqrt(
3))
[sqrt(2),sqrt(3)]
1
EqualComAss
rationalized(1/(1+i))
[i]
1
EqualComAss
rationalized(1/(1+1/root(3,2))
)
[root(3,2)]
1
Differential Equations
EqualComAss
diff(y,x)
0
1
EqualComAss
diff(x^2,x)
2*x
1
EqualComAss
noundiff(x^2,x)
2*x
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
diff(y,x)
'diff(y,x)
0 ATEqualComAss (AlgEquiv-true).
EqualComAss
noundiff(y,x)
'diff(y,x)
1
EqualComAss
'diff(y(x),x)
'diff(y(x),x,1)
1
EqualComAss
noundiff(y(x),x)=-x/4
4*noundiff(y(x),x)+x=0
0 ATEqualComAss (AlgEquiv-true).

## EqualComAssRules

Test
?
Student response
Opt
Mark
CAS errors
Feedback
EqualComAssRules
1/0
0
[]
-1 ATEqualComAssRules_STACKERROR_SAns.
EqualComAssRules
0
1/0
[]
-1 ATEqualComAssRules_STACKERROR_TAns.
EqualComAssRules
0+a
a
EqualComAssRules
0+a
a
x
EqualComAssRules
0+a
a
[x]
EqualComAssRules
0+a
a
[intMul,intFac]
Basic cases
EqualComAssRules
1+1
3
[zeroAdd]
0 ATEqualComAssRules (AlgEquiv-false).
EqualComAssRules
1+1
2
[zeroAdd]
0
EqualComAssRules
1+1
2
[testdebug,zero
Add]
EqualComAssRules
0+a
a
[zeroAdd]
1
EqualComAssRules
a+0
a
[zeroAdd]
1
EqualComAssRules
1*a
a
[testdebug,zero
Add]
0 ATEqualComAssRules: [1 nounmul a,a].
EqualComAssRules
1*a
a
[oneMul]
1
EqualComAssRules
1*a
a
ID_TRANS
1
EqualComAssRules
a/1
a
ID_TRANS
1
EqualComAssRules
0*a
0
ID_TRANS
1
EqualComAssRules
0-1*i
-i
ID_TRANS
1
EqualComAssRules
0-i
-i
ID_TRANS
1
EqualComAssRules
2+1*i
2+i
ID_TRANS
1
EqualComAssRules
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
EqualComAssRules
%e^x
exp(x)
[testdebug,ID_T
RANS]
1 ATEqualComAssRules: [%e nounpow x,%e nounpow x].
EqualComAssRules
12*%e^((2*(%pi/2)*%i)/2)
12*exp(%i*(%pi/2))
ID_TRANS
0
EqualComAssRules
12*%e^((2*(%pi/2)*%i)/2)
12*exp(%i*(%pi/2))
[ID_TRANS,[negN
eg,negDiv,negOr
d],[recipMul,di
vDiv,divCancel]
,intPow]]
1
EqualComAssRules
0^(1-1)
0
ID_TRANS
0 ATEqualComAssRules_STACKERROR_SAns.
EqualComAssRules
0*a
0
delete(zeroMul,
ID_TRANS)
0
EqualComAssRules
-(-a)
a
[negNeg]
1
EqualComAssRules
-(-(-a))
-a
[negNeg]
1
EqualComAssRules
-(-(-a))
a
[testdebug,negN
eg]
0 ATEqualComAssRules (AlgEquiv-false).
EqualComAssRules
3/(-x)
-3/x
ID_TRANS
0
EqualComAssRules
3/(-x)
-3/x
[testdebug,ID_T
RANS]
0 ATEqualComAssRules: [3 nounmul UNARY_RECIP UNARY_MINUS nounmul x,UNARY_MINUS nounmul 3 nounmul UNARY_RECIP x].
EqualComAssRules
-x*(x+1)
x*(-x-1)
[negDist]
1
EqualComAssRules
-x*(x-1)
x*(1-x)
NEG_TRANS
1
EqualComAssRules
-x*(x-1)
x*(1-x)
NEG_TRANS
1
EqualComAssRules
-5*x*(3-x)
5*x*(x-3)
NEG_TRANS
1
EqualComAssRules
-x*(x-1)*(x+1)
x*(x-1)*(-x-1)
NEG_TRANS
1
EqualComAssRules
-x*(x-1)*(x+1)
x*(1-x)*(x+1)
NEG_TRANS
1
EqualComAssRules
-x*(y-1)*(x-1)
x*(1-x)*(y-1)
NEG_TRANS
1
EqualComAssRules
-x*(y-1)*(x-1)
x*(x-1)*(1-y)
NEG_TRANS
1
EqualComAssRules
(x-y)*(y-x)
-(x-y)*(x-y)
NEG_TRANS
1
EqualComAssRules
(x-y)*(y-x)
-(x-y)^2
[testdebug,NEG_
TRANS]
0 ATEqualComAssRules: [UNARY_MINUS nounmul (x nounadd UNARY_MINUS nounmul y) nounmul (x nounadd UNARY_MINUS nounmul y),UNARY_MINUS nounmul (x nounadd UNARY_MINUS nounmul y) nounpow 2].
EqualComAssRules
-x*(x-1)*(x+1)
x*(1-x)*(x+1)
[testdebug,negD
ist,negNeg]
0 ATEqualComAssRules: [x nounmul (UNARY_MINUS nounmul 1 nounadd UNARY_MINUS nounmul x) nounmul (x nounadd UNARY_MINUS nounmul 1),x nounmul (1 nounadd UNARY_MINUS nounmul x) nounmul (1 nounadd x)].
EqualComAssRules
-x*(y-1)*(x-1)
x*(x-1)*(1-y)
[testdebug,negD
ist,negNeg]
0 ATEqualComAssRules: [x nounmul (1 nounadd UNARY_MINUS nounmul x) nounmul (y nounadd UNARY_MINUS nounmul 1),x nounmul (1 nounadd UNARY_MINUS nounmul y) nounmul (x nounadd UNARY_MINUS nounmul 1)].
EqualComAssRules
3/(-x)
-3/x
[negDiv]
1
EqualComAssRules
3/(-x)
ev(-3,simp)/x
[negDiv]
1
EqualComAssRules
(-a)/(-x)
-(-a/x)
[testdebug,ID_T
RANS]
0 ATEqualComAssRules: [UNARY_MINUS nounmul a nounmul UNARY_RECIP UNARY_MINUS nounmul x,UNARY_MINUS nounmul UNARY_MINUS nounmul a nounmul UNARY_RECIP x].
EqualComAssRules
(-a)/(-x)
-(-a/x)
[negDiv]
1
EqualComAssRules
(-a)/(-x)
a/x
[testdebug,negD
iv]
0 ATEqualComAssRules: [UNARY_MINUS nounmul UNARY_MINUS nounmul a nounmul UNARY_RECIP x,a nounmul UNARY_RECIP x].
EqualComAssRules
(-a)/(-x)
a/x
[negDiv,negNeg]
1
EqualComAssRules
1/(-x)
(-1)/x
[negDiv]
1
EqualComAssRules
1/(-x)
ev(-1,simp)/x
[negDiv]
1
EqualComAssRules
(2/-3)*(x-y)
-(2/3)*(x-y)
[negDiv]
1
EqualComAssRules
(2/-3)*(x-y)
(2/3)*(y-x)
[negDiv]
0
EqualComAssRules
(2/-3)*(x-y)
(2/3)*(y-x)
[negDiv,negOrd]
1
EqualComAssRules
-2/(1-x)
2/(x-1)
[testdebug,negD
iv]
0 ATEqualComAssRules: [UNARY_MINUS nounmul 2 nounmul UNARY_RECIP (1 nounadd UNARY_MINUS nounmul x),2 nounmul UNARY_RECIP (x nounadd UNARY_MINUS nounmul 1)].
EqualComAssRules
1/2*3/x
3/(2*x)
[testdebug,ID_T
RANS]
0 ATEqualComAssRules: [3 nounmul (UNARY_RECIP 2) nounmul UNARY_RECIP x,3 nounmul UNARY_RECIP 2 nounmul x].
EqualComAssRules
1/2*3/x
3/(2*x)
[ID_TRANS,recip
Mul]
1
EqualComAssRules
5/2*3/x
15/(2*x)
[testdebug,ID_T
RANS,recipMul]
0 ATEqualComAssRules: [3 nounmul 5 nounmul UNARY_RECIP 2 nounmul x,15 nounmul UNARY_RECIP 2 nounmul x].
EqualComAssRules
-(x-y)
y-x
[negOrd]
1
EqualComAssRules
5/2*3/x
15/(2*x)
[ID_TRANS,recip
Mul,intMul]
1
EqualComAssRules
(3+2)*x+x
5*x+x
[ID_TRANS,intAd
d]
1
EqualComAssRules
(3-5)*x+x
-2*x+x
[ID_TRANS,intAd
d]
1
EqualComAssRules
7*x*(-3*x)
-21*x*x
[ID_TRANS,intMu
l]
1
EqualComAssRules
(-7*x)*(-3*x)
21*x*x
[testdebug,ID_T
RANS,intMul]
0 ATEqualComAssRules: [UNARY_MINUS nounmul UNARY_MINUS nounmul 21 nounmul x nounmul x,21 nounmul x nounmul x].
EqualComAssRules
(-7*x)*(-3*x)
21*x*x
[ID_TRANS,intMu
l,negNeg]
1
ev(a/b/c, simp)=a/(b*c)
EqualComAssRules
a/b/c
a/(b*c)
[testdebug,ID_T
RANS]
0 ATEqualComAssRules: [a nounmul (UNARY_RECIP b) nounmul UNARY_RECIP c,a nounmul UNARY_RECIP b nounmul c].
EqualComAssRules
a/b/c
a/(b*c)
[ID_TRANS,recip
Mul]
1
EqualComAssRules
(a/b)/c
a/(b*c)
[ID_TRANS,recip
Mul]
1
ev(a/(b/c), simp)=(a*c)/b
EqualComAssRules
a/(b/c)
(a*c)/b
[testdebug,ID_T
RANS]
0 ATEqualComAssRules: [a nounmul UNARY_RECIP b nounmul UNARY_RECIP c,a nounmul c nounmul UNARY_RECIP b].
EqualComAssRules
a/(b/c)
(a*c)/b
[testdebug,ID_T
RANS,recipMul]
0 ATEqualComAssRules: [a nounmul UNARY_RECIP b nounmul UNARY_RECIP c,a nounmul c nounmul UNARY_RECIP b].
EqualComAssRules
a/(b/c)
(a*c)/b
[ID_TRANS,divDi
v]
1
EqualComAssRules
A*a/(B*b/c)
A*(a*c)/(B*b)
[ID_TRANS,divDi
v]
1
EqualComAssRules
A*a/(B*b/c)*1/d
A*(a*c)/(B*b)*1/d
[ID_TRANS,divDi
v]
1
EqualComAssRules
D*A*a/(B*b/c)*1/d
A*(a*c)/(B*b)*D/d
[ID_TRANS,divDi
v]
1
EqualComAssRules
A*a/(B*b/c)*1/d
A*(a*c)/(B*b*d)
[testdebug,ID_T
RANS,divDiv]
0 ATEqualComAssRules: [A nounmul a nounmul c nounmul (UNARY_RECIP B nounmul b) nounmul UNARY_RECIP d,A nounmul a nounmul c nounmul UNARY_RECIP B nounmul b nounmul d].
EqualComAssRules
A*a/(B*b/c)*1/d
A*(a*c)/(B*b*d)
[ID_TRANS,divDi
v,recipMul]
1
EqualComAssRules
A/(B/(C/D))
A*C/(B*D)
[testdebug,ID_T
RANS,divDiv]
0 ATEqualComAssRules: [A nounmul C nounmul (UNARY_RECIP B) nounmul UNARY_RECIP D,A nounmul C nounmul UNARY_RECIP B nounmul D].
EqualComAssRules
A/(B/(C/D))
A*C/(B*D)
[ID_TRANS,divDi
v,recipMul]
1
EqualComAssRules
18
2*3^2
[intFac]
1
EqualComAssRules
0+%i*(-(1/27))
-(%i/27)
[[zeroAdd,zeroM
ul,oneMul,onePo
w,idPow,zeroPow
,zPow,oneDiv],[
negNeg,negDiv,n
egOrd],[recipMu
l,divDiv,divCan
tMul,intPow]]
1
EqualComAssRules
x=sqrt(3)+2
x=3^(1/2)+2
[ID_TRANS,sqrtR
em]
1
EqualComAssRules
x=sqrt(3)+2 nounor x=-sqrt(3)-
2
x=3^(1/2)+2 nounor x=-3^(1/2)-
2
ID_TRANS
0
EqualComAssRules
x=sqrt(3)+2 nounor x=-sqrt(3)-
2
x=3^(1/2)+2 nounor x=-3^(1/2)-
2
[ID_TRANS,sqrtR
em]
1
EqualComAssRules
x=sqrt(3)+2 nounor x=-sqrt(3)+
7
x=3^(1/2)+2 nounor x=-3^(1/2)-
2
[ID_TRANS,sqrtR
em]
0 ATEqualComAssRules (AlgEquiv-false)ATEquation_default.
EqualComAssRules
1/sqrt(3)
1/3^(1/2)
[ID_TRANS,sqrtR
em]
1
EqualComAssRules
1/sqrt(3)
3^(-1/2)
[ID_TRANS,sqrtR
em]
0

## CasEqual

Test
?
Student response
Opt
Mark
CAS errors
Feedback
CasEqual
1/0
x^2-2*x+1
-1 ATCASEqual_STACKERROR_SAns.
CasEqual
x
1/0
-1 ATCASEqual_STACKERROR_TAns.
CasEqual
0.5
1/2
x
0 ATCASEqual (AlgEquiv-true).
CasEqual
x=1
1
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
a
A
0 ATCASEqual_false.
CasEqual
exdowncase(X^2-2*X+1)
x^2-2*x+1
1 ATCASEqual_true.
Numbers
CasEqual
4^(-1/2)
1/2
0 ATCASEqual (AlgEquiv-true).
CasEqual
ev(4^(-1/2),simp)
ev(1/2,simp)
1 ATCASEqual_true.
CasEqual
2^2
4
0 ATCASEqual (AlgEquiv-true).
Powers
CasEqual
a^2/b^3
a^2*b^(-3)
0 ATCASEqual (AlgEquiv-true).
Expressions with subscripts
CasEqual
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 ATCASEqual_true.
CasEqual
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 ATCASEqual_false.
Mix of floats and rational numbers
CasEqual
0.5
1/2
0 ATCASEqual (AlgEquiv-true).
CasEqual
x^(1/2)
sqrt(x)
0 ATCASEqual (AlgEquiv-true).
CasEqual
ev(x^(1/2),simp)
ev(sqrt(x),simp)
1 ATCASEqual_true.
CasEqual
abs(x)
sqrt(x^2)
0 ATCASEqual (AlgEquiv-true).
CasEqual
ev(abs(x),simp)
ev(sqrt(x^2),simp)
1 ATCASEqual_true.
CasEqual
x-1
(x^2-1)/(x+1)
0 ATCASEqual (AlgEquiv-true).
Polynomials and rational function
CasEqual
x+x
2*x
0 ATCASEqual (AlgEquiv-true).
CasEqual
ev(x+x,simp)
ev(2*x,simp)
1 ATCASEqual_true.
CasEqual
x+x^2
x^2+x
0 ATCASEqual (AlgEquiv-true).
CasEqual
ev(x+x^2,simp)
ev(x^2+x,simp)
1 ATCASEqual_true.
CasEqual
(x-1)^2
x^2-2*x+1
0 ATCASEqual (AlgEquiv-true).
CasEqual
(x-1)^(-2)
1/(x^2-2*x+1)
0 ATCASEqual (AlgEquiv-true).
CasEqual
1/n-1/(n+1)
1/(n*(n+1))
0 ATCASEqual (AlgEquiv-true).
Trig functions
CasEqual
cos(x)
cos(-x)
0 ATCASEqual (AlgEquiv-true).
CasEqual
ev(cos(x),simp)
ev(cos(-x),simp)
1 ATCASEqual_true.
CasEqual
cos(x)^2+sin(x)^2
1
0 ATCASEqual (AlgEquiv-true).
CasEqual
2*cos(x)^2-1
cos(2*x)
0 ATCASEqual (AlgEquiv-true).
Predicate function wrapper
CasEqual
imag_numberp(2*%i)
true
1 ATCASEqual_true.
CasEqual
imag_numberp(%e^(%i*%pi/2))
true
1 ATCASEqual_true.
CasEqual
imag_numberp(2)
false
1 ATCASEqual_true.
CasEqual
imag_numberp(%e^(%pi/2))
false
1 ATCASEqual_true.
CasEqual
complex_exponentialp(3*%e^(%i*
%pi/6))
true
1 ATCASEqual_true.
CasEqual
complex_exponentialp(3)
true
1 ATCASEqual_true.
CasEqual
complex_exponentialp(%e^(%i*%p
i/6))
true
1 ATCASEqual_true.
CasEqual
complex_exponentialp(%e^%i)
true
1 ATCASEqual_true.
CasEqual
complex_exponentialp(%e^(%pi/6
))
true
1 ATCASEqual_true.
CasEqual
complex_exponentialp(3+%i)
false
1 ATCASEqual_true.
CasEqual
complex_exponentialp(%e^(%i)/4
)
true
1 ATCASEqual_true.
CasEqual
complex_exponentialp(3*exp(%i*
%pi/6))
true
1 ATCASEqual_true.
CasEqual
integerp(-1)
true
0 ATCASEqual_false.
CasEqual
integerp(ev(-1,simp))
true
1 ATCASEqual_true.

## SameType

Test
?
Student response
Opt
Mark
CAS errors
Feedback
SameType
1/0
1
SameType
1
1/0
Numbers
SameType
4^(-1/2)
1/2
1
Lists
SameType
x
[1,2,3]
0
SameType
[1,2]
[1,2,3]
1
SameType
[1,x>2]
[1,2<x]
1
SameType
[1,x,3]
[1,2<x,4]
0
Sets
SameType
x
{1,2,3}
0
SameType
{1,2}
{1,2,3}
1
Matrices
SameType
matrix([1,2],[2,3])
matrix([1,2],[2,3])
1
SameType
[[1,2],[2,3]]
matrix([1,2],[2,3])
0
SameType
matrix([1,2],[2,3])
matrix([1,2,3],[2,3,3])
1
SameType
matrix([x>4,{1,x^2}],[[1,2]
,[1,3]])
matrix([4-x<0,{x^2, 1}],[[1
,2],[1,3]])
1
SameType
matrix([x>4,[1,x^2]],[[1,2]
,[1,3]])
matrix([4-x<0,{x^2, 1}],[[1
,2],[1,4]])
0
Equations
SameType
1
x=1
0
SameType
x=1
x=1
1
Inequalities
SameType
1
x>1
0
SameType
x>2
x>1
1
SameType
x>1
x>=1
1
SameType
x>1 and x<3
x>=1
1
SameType
{x>1,x<3}
x>=1
0
SameType
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

## SysEquiv

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Basic tests
SysEquiv
1/0
[(x-1)*(x+1)=0]
SysEquiv
[(x-1)*(x+1)=0]
1/0
SysEquiv
1
[(x-1)*(x+1)=0]
0 Your answer should be a list, but it is not! ATSysEquiv_SA_not_list.
SysEquiv
[(x-1)*(x+1)=0]
1
SysEquiv
[1]
[90=v*t,90=(v+5)*(t-1/4)]
0 Your answer should be a list of equations, but it is not! ATSysEquiv_SA_not_eq_list.
SysEquiv
[(x-1)*(x+1)=0]
[1]
0 The teacher's answer is not a list of equations, but should be. ATSysEquiv_SB_not_eq_list.
SysEquiv
[x^2]
[(x-1)*(x+1)=0]
0 Your answer should be a list of equations, but it is not! ATSysEquiv_SA_not_eq_list.
SysEquiv
[90=v*t^t,90=(v+5)*(t-1/4)]
[90=v*t,90=(v+5)*(t-1/4)]
0 One or more of your equations is not a polynomial! ATSysEquiv_SA_not_poly_eq_list.
SysEquiv
[90=v*t,90=(v+5)*(t-1/4)]
[90=v*t^t,90=(v+5)*(t-1/4)]
Tests of equivalence
SysEquiv
[x^2=1]
[(x-1)*(x+1)=0]
1
SysEquiv
[x^2+y^2=4,y=x]
[y=x,y^2=2]
1
SysEquiv
[x^2+y^2=2,y=x]
[y=x,y^2=2]
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
[x=1]
[(x-1)*(x+1)=0,(x-1)*(x-3)=0]
1 ATSysEquiv_SA_Completely_solved.
SysEquiv
[3*a+b-c=2, a-b+2*c=5,b+c=5]
[a=1,b=2,c=3]
1
SysEquiv
[a=1,b=2,c=3]
[3*a+b-c=2, a-b+2*c=5,b+c=5]
1 ATSysEquiv_SA_Completely_solved.
SysEquiv
[x^2=1]
[(x-1)*(x+1)*(x-2)=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
[x=1,y=-1]
[(x-1)*(y+1)=0]
0 ATSysEquiv_SA_Not_completely_solved.
SysEquiv
[x=1]
[(x-1)*(x+1)=0]
0 ATSysEquiv_SA_Not_completely_solved.
SysEquiv
[x=1]
[(x-1)*(x+1)*y=0]
0 ATSysEquiv_SA_Not_completely_solved.
SysEquiv
[90=v*t,90=(v+5)*(t-1/4)]
[90=v*t,90=(v+5)*(t-1/4)]
1
SysEquiv
[90=v*t,90=(v+5)*(t*x-1/4)]
[90=v*t,90=(v+5)*(t-1/4)]
SysEquiv
[90=v*t,90=(v+5)*(t-1/4)]
[90=v*t,90=(v+5)*(t*x-1/4)]
SysEquiv
[90=v*t]
[90=v*t,90=(v+5)*(t-1/4)]
0 The equations in your system appear to be correct, but you need others besides. ATSysEquiv_SA_system_underdetermined.
SysEquiv
[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 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
[90=v*t,90=(v+5)*(t-1/4),90=(v
+6)*(t-1/5),90=(v+7)*(t-1/4),9
0=(v+8)*(t-1/3)]
[90=v*t,90=(v+5)*(t-1/4)]
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
[b^2=a,a=9]
[x^2=y,y=9]
SysEquiv
[x^2=4]
[x^2=4,y=9]
SysEquiv
[d=90,d=v*t,d=(v+5)*(t-1/4)]
[90=v*t,90=(v+5)*(t-1/4)]
SysEquiv
stack_eval_assignments([d=90,d
=v*t,d=(v+5)*(t-1/4)])
[90=v*t,90=(v+5)*(t-1/4)]
1

## Sets

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Sets
{1/0}
{0}
Sets
{0}
{1/0}
Sets
x
{1,2,3}
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
{1,2}
x
Sets
{1,2}
{1,2,3}
0 The following are missing from your set. $\left \{3 \right \}$ ATSets_missingentries.
Sets
{1,2,4}
{1,2}
0 These entries should not be elements of your set. $\left \{4 \right \}$ ATSets_wrongentries.
Sets
{1,2,2+2}
{1,2}
0 These entries should not be elements of your set. $\left \{4 \right \}$ ATSets_wrongentries.
Sets
{5,1,2,4}
{1,2,3}
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
{2/4, 1/3}
{1/2, 1/3}
1
Duplicate entries
Sets
{1,2,1}
{1,2}
1 Your set appears to contain duplicate entries! ATSets_duplicates.
Sets
{1,2,1+1}
{1,2}
1 Your set appears to contain duplicate entries! ATSets_duplicates.
Sets
{1,2,1+1}
{1,2,3}
0 Your set appears to contain duplicate entries! The following are missing from your set. $\left \{3 \right \}$ ATSets_duplicates. ATSets_missingentries.
Sets
{(x-a)^6000}
{(a-x)^6000}
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

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Expanded
1/0
0
Expanded
x>2
x^2-2*x+1
0 Your answer should be an expression, not an equation, inequality, list, set or matrix. ATExpanded_SA_not_expression.
Expanded
x^2-1
0
1 ATExpanded_TRUE.
Expanded
2*(x-1)
0
0 ATExpanded_FALSE.
Expanded
(x-1)*(x+1)
0
0 ATExpanded_FALSE.
Expanded
(x-a)*(x-b)
0
0 ATExpanded_FALSE.
Expanded
x^2-(a+b)*x+a*b
0
0 ATExpanded_FALSE.
Expanded
x^2-a*x-b*x+a*b
0
1 ATExpanded_TRUE.
Expanded
cos(2*x)
0
1 ATExpanded_TRUE.
Expanded
p+1
0
1 ATExpanded_TRUE.
Expanded
(p+1)*(p-1)
0
0 ATExpanded_FALSE.
Expanded
3+2*sqrt(3)
0
1 ATExpanded_TRUE.
Expanded
3+sqrt(12)
0
1 ATExpanded_TRUE.
Expanded
(1+sqrt(5))*(1-sqrt(3))
0
0 ATExpanded_FALSE.
This fails, but you are never going to ask students to do this anyway...
Expanded !
(a-x)^6000
0
-2 ATExpanded_TRUE.

## FacForm

Test
?
Student response
Opt
Mark
CAS errors
Feedback
FacForm
1/0
0
x
-1 ATFacForm_STACKERROR_SAns.
FacForm
0
1/0
x
-1 ATFacForm_STACKERROR_TAns.
FacForm
0
0
1/0
-1 ATFacForm_STACKERROR_Opt.
Trivial cases
FacForm
2
2
x
1 ATFacForm_int_true.
FacForm
6
6
x
1 ATFacForm_int_true.
FacForm
1/3
1/3
x
1 ATFacForm_true.
FacForm
3*x^2
3*x^2
x
1 ATFacForm_true.
FacForm
4*x^2
4*x^2
x
1 ATFacForm_true.
Linear integer factors
FacForm
2*(x-1)
2*x-2
x
1 ATFacForm_true.
FacForm
2*x-2
2*x-2
x
0 Your answer is not factored. You need to take out a common factor. ATFacForm_notfactored.
FacForm
2*(x+1)
2*x-2
x
FacForm
2*x+2
2*x-2
x
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
2*(x+0.5)
2*x+1
x
1 ATFacForm_default_true.
Linear factors
FacForm
t*(2*x+1)
t*(2*x+1)
x
1 ATFacForm_true.
FacForm
t*x+t
t*(x+1)
x
FacForm
6*s*t+10*s
2*s*(3*t+5)
t
FacForm
2*x*(x-3)
2*x^2-6*x
x
1 ATFacForm_true.
FacForm
2*(x^2-3*x)
2*x*(x-3)
x
0 Your answer is not factored. You could still do some more work on the term $$x^2-3\cdot x$$. ATFacForm_notfactored.
FacForm
x*(2*x-6)
2*x*(x-3)
x
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
(x+2)*(x+3)
(x+2)*(x+3)
x
1 ATFacForm_true.
FacForm
(x+2)*(2*x+6)
2*(x+2)*(x+3)
x
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
(z*x+z)*(2*x+6)
2*z*(x+1)*(x+3)
x
0 Your answer is not factored. You could still do some more work on the term $$z\cdot x+z$$. 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
(x+t)*(x-t)
x^2-t^2
x
1 ATFacForm_true.
FacForm
t^2-1
(t-1)*(t+1)
t
FacForm
t^2+1
t^2+1
t
1 ATFacForm_true.
FacForm
v^2+1
v^2+1
v
1 ATFacForm_true.
FacForm
v^2-1
v^2-1
v
FacForm
-(3*w-4*v+9*u)*(3*w+4*v-u)
-(3*w-4*v+9*u)*(3*w+4*v-u)
v
1 ATFacForm_true.
FacForm
-6*k*(4*b-k-1)
6*k*(1+k-4*b)
k
1 ATFacForm_default_true.
FacForm
-2*3*k*(4*b-k-1)
6*k*(1+k-4*b)
k
1 ATFacForm_true.
FacForm
-(6*k*(4*b-k-1))
6*k*(1+k-4*b)
k
1 ATFacForm_default_true.
FacForm
-(6*a*(4*b-a-1))
6*a*(1+a-4*b)
a
1 ATFacForm_true.
FacForm
-(6*a*(4*b-a-1))
6*a*(-(4*b)+a+1)
a
1 ATFacForm_true.
FacForm
x*(x-4+4/x)
x^2-4*x+4
x
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
(2-x)*(3-x)
(x-2)*(x-3)
x
1 ATFacForm_true.
FacForm
(1-x)^2
(x-1)^2
x
1 ATFacForm_true.
FacForm
(1-x)*(1-x)
(x-1)^2
x
1 ATFacForm_true.
FacForm
-(1-x)^2
-(x-1)^2
x
1 ATFacForm_true.
FacForm
(1-x)^2
(x-1)^2
x
1 ATFacForm_true.
FacForm
4*(1-x/2)^2
(x-2)^2
x
1 ATFacForm_default_true.
FacForm
-3*(x-4)*(x+1)
-3*x^2+9*x+12
x
1 ATFacForm_true.
FacForm
3*(-x+4)*(x+1)
-3*x^2+9*x+12
x
1 ATFacForm_true.
FacForm
3*(4-x)*(x+1)
-3*x^2+9*x+12
x
1 ATFacForm_true.
Cubics
FacForm
(x-1)*(x^2+x+1)
x^3-1
x
1 ATFacForm_true.
FacForm
x^3-x+1
x^3-x+1
x
1 ATFacForm_true.
FacForm
7*x^3-7*x+7
7*(x^3-x+1)
x
0 Your answer is not factored. You need to take out a common factor. ATFacForm_notfactored.
FacForm
(1-x)*(2-x)*(3-x)
-x^3+6*x^2-11*x+6
x
1 ATFacForm_true.
FacForm
(2-x)*(2-x)*(3-x)
-x^3+7*x^2-16*x+12
x
1 ATFacForm_true.
FacForm
(2-x)^2*(3-x)
-x^3+7*x^2-16*x+12
x
1 ATFacForm_true.
FacForm
(x^2-4*x+4)*(3-x)
-x^3+7*x^2-16*x+12
x
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
(x^2-3*x+2)*(3-x)
-x^3+6*x^2-11*x+6
x
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
3*y^3-6*y^2-24*y
3*(y-4)*y*(y+2)
y
0 Your answer is not factored. You need to take out a common factor. ATFacForm_notfactored.
FacForm
3*(y^3-2*y^2-8*y)
3*(y-4)*y*(y+2)
y
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
3*y*(y^2-2*y-8)
3*(y-4)*y*(y+2)
y
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
3*(y^2-4*y)*(y+2)
3*(y-4)*y*(y+2)
y
0 Your answer is not factored. You could still do some more work on the term $$y^2-4\cdot y$$. ATFacForm_notfactored.
FacForm
(y-4)*y*(3*y+6)
3*(y-4)*y*(y+2)
y
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
(a-x)^6000
(a-x)^6000
x
1 ATFacForm_true.
FacForm
(x-a)^6000
(a-x)^6000
x
1 ATFacForm_true.
Needs flattening
FacForm
2*a*(a*b-1)
2*a*(a*b-1)
a
1 ATFacForm_true.
FacForm
(2*a)*(a*b-1)
2*a*(a*b-1)
a
1 ATFacForm_true.
FacForm
3*x*(7*y-3)*(7*y+3)
3*x*(7*y-3)*(7*y+3)
x
1 ATFacForm_true.
FacForm
3*x*(7*y-3)*(7*y+3)
3*x*(7*y-3)*(7*y+3)
y
1 ATFacForm_true.
Not polynomials in a variable
FacForm
(sin(x)+1)*(sin(x)-1)
sin(x)^2-1
sin(x)
1 ATFacForm_true.
FacForm
(cos(t)-sqrt(2))^2
cos(t)^2-2*sqrt(2)*cos(t)+2
cos(t)
1 ATFacForm_true.
FacForm
7
7
x
1 ATFacForm_int_true.
Factors over other fields
FacForm
24*(x-1/4)
24*x-6
x
1 ATFacForm_default_true.
FacForm
(x-sqrt(2))*(x+sqrt(2))
x^2-2
x
1 ATFacForm_true.
FacForm
x^2-2
x^2-2
x
1 ATFacForm_true.
FacForm
(%i*x-2*%i)
%i*(x-2)
x
FacForm
%i*(x-2)
(%i*x-2*%i)
x
1 ATFacForm_true.
FacForm
(x-%i)*(x+%i)
x^2+1
x
1 ATFacForm_true.
FacForm
(x-1)*(x+(1+sqrt(3)*%i)/2)*(x+
(1-sqrt(3)*%i)/2)
x^3-1
x
1 ATFacForm_default_true.

## CompSquare

Test
?
Student response
Opt
Mark
CAS errors
Feedback
CompSquare
1/0
0
CompSquare
1/0
0
x
CompSquare
0
1/0
x
CompSquare
0
0
1/0
Category errors.
CompSquare
1
(x-1)^2+1
x
0 Your answer should depend on the variable $$x$$ but it does not! ATCompSquare_SA_not_depend_var.
CompSquare
(t-1)^2+1
(x-1)^2+1
x
0 Your answer should depend on the variable $$x$$ but it does not! ATCompSquare_SA_not_depend_var.
CompSquare
(x-1)^2+1=0
(x-1)^2+1
x
0 Your answer should be an expression, not an equation, inequality, list, set or matrix. ATCompSquare_STACKERROR_LIST.
CompSquare
sin(x-1)+a-1
(x-1)^2+1
x
0 ATCompSquare_false_not_AlgEquiv.
Trivial cases
CompSquare
1
1
x
1 ATCompSquare_true_trivial.
CompSquare
x-a
x-a
x
1 ATCompSquare_true_trivial.
CompSquare
x^2
x^2
x
1 ATCompSquare_true.
CompSquare
x^2-1
(x-1)*(x+1)
x
1 ATCompSquare_true.
CompSquare
(x-1)^2*k
(x-1)^2*k
x
1 ATCompSquare_true.
CompSquare
(x-1)^2/k
(x-1)^2/k
x
1 ATCompSquare_true.
Normal cases
CompSquare
(x-1)^2+1
(x-1)^2+1
x
1 ATCompSquare_true.
CompSquare
(1-x)^2+1
(x-1)^2+1
x
1 ATCompSquare_true.
CompSquare
(X-1)^2+1
(x-1)^2+1
x
0 Your answer should depend on the variable $$x$$ but it does not! ATCompSquare_SA_not_depend_var.
CompSquare
9*(x-1)^2+1
(3*x-3)^2+1
x
1 ATCompSquare_true.
CompSquare
-(x-1)^2
-(x-1)^2
x
1 ATCompSquare_true.
CompSquare
-(1-x)^2
-(x-1)^2
x
1 ATCompSquare_true.
CompSquare
-(x-1)^2+3
-(x-1)^2+3
x
1 ATCompSquare_true.
CompSquare
-(1-x)^2+3
-(x-1)^2+3
x
1 ATCompSquare_true.
CompSquare
-4*(x-1)^2+3
-4*(x-1)^2+3
x
1 ATCompSquare_true.
CompSquare
-4*(x-1)^2+3
-(2*x-2)^2+3
x
1 ATCompSquare_true.
CompSquare
3-4*(x-1)^2
-(2*x-2)^2+3
x
1 ATCompSquare_true.
CompSquare
(x-1)^2+1
(x+1)^2+1
x
0 Your answer appears to be in the correct form, but is not equivalent to the correct answer. ATCompSquare_true_not_AlgEquiv.
CompSquare
(x-a^2)^2+1+b
(x-a^2)^2+1+b
x
1 ATCompSquare_true.
CompSquare
x^2-2*x+2
(x-1)^2+1
x
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
x+1
(x-1)^2+1
x
0 ATCompSquare_false_not_AlgEquiv.
CompSquare
a*(x-1)^2+1
a*(x-1)^2+1
x
1 ATCompSquare_true.
CompSquare
-a*(x-1)^2+1
1-a*(x-1)^2
x
1 ATCompSquare_true.
Not simple variable
CompSquare
(sin(x)-1)^2+1
(sin(x)-1)^2+1
sin(x)
1 ATCompSquare_true.
CompSquare
(x^2-1)^2+1
(x^2-1)^2+1
x^2
1 ATCompSquare_true.
CompSquare
(y-1)^2+1
(y-1)^2+1
y
1 ATCompSquare_true.
CompSquare
(y+1)^2+1
(y-1)^2+1
y
0 Your answer appears to be in the correct form, but is not equivalent to the correct answer. ATCompSquare_true_not_AlgEquiv.
CompSquare
(x-1)^2+1
(sin(x)-1)^2+1
sin(x)
0 Your answer should depend on the variable $${\it facdum}$$ but it does not! ATCompSquare_SA_not_depend_var.

## PropLogic

Test
?
Student response
Opt
Mark
CAS errors
Feedback
PropLogic
1/0
0
PropLogic
0
1/0
PropLogic
true
true
1
PropLogic
true
false
0
PropLogic
A implies B
not(A) or B
1
PropLogic
(a and b and c) xor (a and b)
xor (a and c) xor a xor true
(a implies b) or c
1

## Equiv

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Equiv
x
[x^2=4,x=2 or x=-2]
Equiv
[x^2=4,x=2 or x=-2]
x
Equiv
[1/0]
[x^2=4,x=2 or x=-2]
-1 ATEquiv_STACKERROR_SAns.
Equiv
[x^2=4,x=2 or x=-2]
[1/0]
-1 ATEquiv_STACKERROR_TAns.
Equiv
[x^2=4,x=2 or x=-2]
[x^2=4,x=2 or x=-2]
1 $\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[x^2=4,x=#pm#2,x=2 and x=-2]
[x^2=4,x=2 or x=-2]
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
[x^2=4,x=2]
[x^2=4,x=2 or x=-2]
0 $\begin{array}{lll} &x^2=4& \cr \color{red}{\Leftarrow}&x=2& \cr \end{array}$ (EMPTYCHAR,IMPLIEDCHAR)
Equiv
[x^2=4,x=2]
[x^2=4,x=2]
[assumepos]
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
[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 $\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x^2-4=0& \cr \color{green}{\Leftrightarrow}&\left(x-2\right)\cdot \left(x+2\right)=0& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[x^2=4,x= #pm#2, x=2 or x=-2]
[x^2=4,x=2 or x=-2]
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
[x^2-6*x+9=0,x=3]
[x^2-6*x+9=0,x=3]
1 $\begin{array}{lll} &x^2-6\cdot x+9=0& \cr \color{green}{\mbox{(Same roots)}}&x=3& \cr \end{array}$ (EMPTYCHAR,SAMEROOTS)
Equiv
[]
[]
1 $\begin{array}{lll} &\left[ \right] & \cr \end{array}$ (EMPTYCHAR)
Equiv
[x^2=-1]
[]
1 $\begin{array}{lll} &x^2=-1& \cr \end{array}$ (EMPTYCHAR)
Equiv
[x=x,all]
[]
1 $\begin{array}{lll} &x=x& \cr \color{green}{\Leftrightarrow}&\mathbb{R}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[x=x,true]
[]
1 $\begin{array}{lll} &x=x& \cr \color{green}{\Leftrightarrow}&\mathbf{True}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[x=x,false]
[]
0 $\begin{array}{lll} &x=x& \cr \color{red}{?}&\mathbf{False}& \cr \end{array}$ (EMPTYCHAR,QMCHAR)
Equiv
[1=1,all]
[]
1 $\begin{array}{lll} &1=1& \cr \color{green}{\Leftrightarrow}&\mathbb{R}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[1=1,true]
[]
1 $\begin{array}{lll} &1=1& \cr \color{green}{\Leftrightarrow}&\mathbf{True}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[0=0,all]
[]
1 $\begin{array}{lll} &0=0& \cr \color{green}{\Leftrightarrow}&\mathbb{R}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[0=0,true]
[]
1 $\begin{array}{lll} &0=0& \cr \color{green}{\Leftrightarrow}&\mathbf{True}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[1=2,false]
[]
1 $\begin{array}{lll} &1=2& \cr \color{green}{\Leftrightarrow}&\mathbf{False}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[1=2,none]
[]
1 $\begin{array}{lll} &1=2& \cr \color{green}{\Leftrightarrow}&\emptyset& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[1=2,{}]
[]
1 $\begin{array}{lll} &1=2& \cr \color{green}{\Leftrightarrow}&\left \{ \right \}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[1=2,[]]
[]
1 $\begin{array}{lll} &1=2& \cr \color{green}{\Leftrightarrow}&\left[ \right] & \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[x=1,X=1]
[]
0 $\begin{array}{lll} &x=1& \cr \color{red}{?}&X=1& \cr \end{array}$ (EMPTYCHAR,QMCHAR)
Equiv
[1/(x^2+1)=1/((x+%i)*(x-%i)),t
rue]
[]
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
[2^2,stackeq(4)]
[]
1 $\begin{array}{lll} &2^2& \cr \color{green}{\checkmark}&=4& \cr \end{array}$ (EMPTYCHAR, CHECKMARK)
Equiv
[2^2,stackeq(3)]
[]
0 $\begin{array}{lll} &2^2& \cr \color{red}{\Rightarrow}&=3& \cr \end{array}$ (EMPTYCHAR,IMPLIESCHAR)
Equiv
[2^2,4]
[]
1 $\begin{array}{lll} &2^2& \cr \color{green}{\Leftrightarrow}&4& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[2^2,3]
[]
0 $\begin{array}{lll} &2^2& \cr \color{red}{\Rightarrow}&3& \cr \end{array}$ (EMPTYCHAR,IMPLIESCHAR)
Equiv
[lg(64,4),lg(4^3,4),3*lg(4,4),
3]
[]
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
[lg(64,4),stackeq(lg(4^3,4)),s
tackeq(3*lg(4,4)),stackeq(3)]
[]
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
[x=1 or x=2,x=1 or 2]
[]
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
[x=1 or x=2,x=1 and x=2]
[]
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
[x=1 and y=2,x=1 or y=2]
[]
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
[a=b,a^2=b^2]
[]
0 $\begin{array}{lll} &a=b& \cr \color{red}{\Rightarrow}&a^2=b^2& \cr \end{array}$ (EMPTYCHAR,IMPLIESCHAR)
Equiv
[a=b,sqrt(a)=sqrt(b)]
[]
0 $\begin{array}{lll} &a=b& \cr \color{red}{\Leftarrow}&\sqrt{a}=\sqrt{b}& \cr \end{array}$ (EMPTYCHAR,IMPLIEDCHAR)
Equiv
[a^2=b^2,a=b]
[]
0 $\begin{array}{lll} &a^2=b^2& \cr \color{red}{\Leftarrow}&a=b& \cr \end{array}$ (EMPTYCHAR,IMPLIEDCHAR)
Equiv
[a^2=b^2,a=b or a=-b]
[]
1 $\begin{array}{lll} &a^2=b^2& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=-b& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[a^2=b^2,a= #pm#b,a= b or a=-b
]
[]
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
[9*x^2/2-81*x/2+90=5*x^2/2-5*x
-20 nounor 9*x^2/2-81*x/2+90=-
(5*x^2/2-5*x-20),9*x^2-81*x+18
0=5*x^2-10*x-40 nounor 9*x^2-8
1*x+180=-5*x^2+10*x+40,4*x^2-7
1*x+220=0 nounor 14*x^2-91*x+1
40=0,x=(71 #pm# sqrt(71^2-4*4*
220))/(2*4) nounor x=(91 #pm#
sqrt(91^2-4*14*140))/(2*14),x=
55/4 nounor x=4 nounor x=5/2]
[]
1 $\begin{array}{lll} &\frac{9\cdot x^2}{2}-\frac{81\cdot x}{2}+90=\frac{5\cdot x^2}{2}-5\cdot x-20\,{\mbox{ or }}\, \frac{9\cdot x^2}{2}-\frac{81\cdot x}{2}+90=-\left(\frac{5\cdot x^2}{2}-5\cdot x-20\right)& \cr \color{green}{\Leftrightarrow}&9\cdot x^2-81\cdot x+180=5\cdot x^2-10\cdot x-40\,{\mbox{ or }}\, 9\cdot x^2-81\cdot x+180=-5\cdot x^2+10\cdot x+40& \cr \color{green}{\Leftrightarrow}&4\cdot x^2-71\cdot x+220=0\,{\mbox{ or }}\, 14\cdot x^2-91\cdot x+140=0& \cr \color{green}{\Leftrightarrow}&x=\frac{{71 \pm \sqrt{71^2-4\cdot 4\cdot 220}}}{2\cdot 4}\,{\mbox{ or }}\, x=\frac{{91 \pm \sqrt{91^2-4\cdot 14\cdot 140}}}{2\cdot 14}& \cr \color{green}{\mbox{(Same roots)}}&x=\frac{55}{4}\,{\mbox{ or }}\, x=4\,{\mbox{ or }}\, x=\frac{5}{2}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,SAMEROOTS)
Equiv
[a=b,abs(a)=abs(b),a=b]
[]
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
[abs(a)=abs(b),a=b or a=-b]
[]
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
[abs(a)=abs(b),a^2=b^2]
[]
1 $\begin{array}{lll} &\left| a\right| =\left| b\right| & \cr \color{green}{\Leftrightarrow}&a^2=b^2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
Equiv
[x^3=8,x=2]
[]
0 $\begin{array}{lll} &x^3=8& \cr \color{red}{\Leftarrow}&x=2& \cr \end{array}$ (EMPTYCHAR,IMPLIEDCHAR)
Equiv
[x^3=8,x=2]
[]
[assumereal]
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
[abs(x-1/2)+abs(x+1/2)=2,abs(x
)=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
[a^2=9 and a>0,a=3]
[]
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
[T=2*pi*sqrt(L/g),T^2=4*pi^2*L
/g,g=4*pi^2*L/T^2]
[]
[assumepos]
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
[a=b,a^2=b^2]
[]
[assumepos]
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
[a=b,sqrt(a)=sqrt(b)]
[]
[assumepos]
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
[a^2=b^2,a=b]
[]
[assumepos]
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
[a^2=b^2,a=b or a=-b]
[]
[assumepos]
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
[a=b,abs(a)=abs(b)]
[]
[assumepos]
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
[abs(a)=abs(b),a=b]
[]
[assumepos]
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
[abs(a)=abs(b),a=-b]
[]
[assumepos]
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
[abs(a)=abs(b),a=b or a=-b]
[]
[assumepos]
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
[x=abs(-2),x=2]
[]
[assumepos]
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
[abs(a)=abs(b),a^2=b^2]
[]
[assumepos]
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
[x^2=9,x=#pm#3,x=3 or x=-3,x=3
]
[]
[assumepos]
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
[x^2=9,x=3]
[]
[assumepos]
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
[x^2=2,x=#pm#sqrt(2),x=sqrt(2)
or x=-sqrt(2)]
[]
[assumepos]
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
[x^2=2,x=sqrt(2)]
[]
[assumepos]
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
[x^2 = a^2-b,x = sqrt(a^2-b)]
[]
[assumepos]
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
[2*(x-3) = 4*x-3*(x+2),2*x-6=x
-6,x=0]
[]
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
[2*(x-3) = 5*x-3*(x+2),2*x-6=2
*x-6,0=0,all]
[]
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
[2*(x-3) = 5*x-3*(x+1),2*x-6=2
*x-3,0=3,{}]
[]
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
[a^2=b^2,a^2-b^2=0,(a-b)*(a+b)
=0,a=b or a=-b]
[]
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
[a^3=b^3,a^3-b^3=0,(a-b)*(a^2+
a*b+b^2)=0,(a-b)=0,a=b]
[]
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
[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 $\begin{array}{lll} &a^3=b^3& \cr \color{green}{\Leftrightarrow}&a^3-b^3=0& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a^2+a\cdot b+b^2\right)=0& \cr \color{green}{\Leftrightarrow}&a-b=0\,{\mbox{ or }}\, a^2+a\cdot b+b^2=0& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, \left(a+\frac{1+\mathrm{i}\cdot \sqrt{3}}{2}\cdot b\right)\cdot \left(a+\frac{1-\mathrm{i}\cdot \sqrt{3}}{2}\cdot b\right)=0& \cr \color{green}{\Leftrightarrow}&a=b\,{\mbox{ or }}\, a=\frac{-\left(1+\mathrm{i}\cdot \sqrt{3}\right)}{2}\cdot b\,{\mbox{ or }}\, a=\frac{-\left(1-\mathrm{i}\cdot \sqrt{3}\right)}{2}\cdot b& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &x^2-x=30& \cr \color{green}{\Leftrightarrow}&x^2-x-30=0& \cr \color{green}{\Leftrightarrow}&\left(x-6\right)\cdot \left(x+5\right)=0& \cr \color{green}{\Leftrightarrow}&x-6=0\,{\mbox{ or }}\, x+5=0& \cr \color{green}{\Leftrightarrow}&x=6\,{\mbox{ or }}\, x=-5& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[x^2=2,x^2-2=0,(x-sqrt(2))*(x+
sqrt(2))=0,x=sqrt(2) or x=-sqr
t(2)]
[]
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
[x^2=2,x=#pm#sqrt(2),x=sqrt(2)
or x=-sqrt(2)]
[]
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
[(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 $\begin{array}{lll} &{\left(2\cdot x-7\right)}^2={\left(x+1\right)}^2& \cr \color{green}{\Leftrightarrow}&{\left(2\cdot x-7\right)}^2-{\left(x+1\right)}^2=0& \cr \color{green}{\Leftrightarrow}&\left(2\cdot x-7+x+1\right)\cdot \left(2\cdot x-7-x-1\right)=0& \cr \color{green}{\Leftrightarrow}&\left(3\cdot x-6\right)\cdot \left(x-8\right)=0& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=8& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[x^2-6*x=-9,(x-3)^2=0,x-3=0,x=
3]
[]
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
[(2*x-7)^2=(x+1)^2,sqrt((2*x-7
)^2)=sqrt((x+1)^2),2*x-7=x+1,x
=8]
[]
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
[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 $\begin{array}{lll} &x^2-10\cdot x+9=0& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2-16=0& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2=16& \cr \color{green}{\Leftrightarrow}&x-5= \pm 4& \cr \color{green}{\Leftrightarrow}&x-5=4\,{\mbox{ or }}\, x-5=-4& \cr \color{green}{\Leftrightarrow}&x=1\,{\mbox{ or }}\, x=9& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &x^2-2\cdot p\cdot x-q=0& \cr \color{green}{\Leftrightarrow}&x^2-2\cdot p\cdot x=q& \cr \color{green}{\Leftrightarrow}&x^2-2\cdot p\cdot x+p^2=q+p^2& \cr \color{green}{\Leftrightarrow}&{\left(x-p\right)}^2=q+p^2& \cr \color{green}{\Leftrightarrow}&x-p= \pm \sqrt{q+p^2}& \cr \color{green}{\Leftrightarrow}&x-p=\sqrt{q+p^2}\,{\mbox{ or }}\, x-p=-\sqrt{q+p^2}& \cr \color{green}{\Leftrightarrow}&x=p+\sqrt{q+p^2}\,{\mbox{ or }}\, x=p-\sqrt{q+p^2}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[x^2-10*x+7=0,(x-5)^2-18=0,(x-
5)^2=sqrt(18)^2,(x-5)^2-sqrt(1
8)^2=0,(x-5-sqrt(18))*(x-5+sqr
t(18))=0,x=5-sqrt(18) or x=5+s
qrt(18)]
[]
1 $\begin{array}{lll} &x^2-10\cdot x+7=0& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2-18=0& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2={\sqrt{18}}^2& \cr \color{green}{\Leftrightarrow}&{\left(x-5\right)}^2-{\sqrt{18}}^2=0& \cr \color{green}{\Leftrightarrow}&\left(x-5-\sqrt{18}\right)\cdot \left(x-5+\sqrt{18}\right)=0& \cr \color{green}{\Leftrightarrow}&x=5-\sqrt{18}\,{\mbox{ or }}\, x=5+\sqrt{18}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[9*x^2/2-81*x/2+90=5*x^2/2-5*x
-20,4*x^2-71*x+220 = 0,x = (71
#pm# 39)/8,x=55/4 nounor x=4]
[]
1 $\begin{array}{lll} &\frac{9\cdot x^2}{2}-\frac{81\cdot x}{2}+90=\frac{5\cdot x^2}{2}-5\cdot x-20& \cr \color{green}{\Leftrightarrow}&4\cdot x^2-71\cdot x+220=0& \cr \color{green}{\Leftrightarrow}&x=\frac{{71 \pm 39}}{8}& \cr \color{green}{\Leftrightarrow}&x=\frac{55}{4}\,{\mbox{ or }}\, x=4& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &x^2+2\cdot a\cdot x=0& \cr \color{green}{\Leftrightarrow}&x\cdot \left(x+2\cdot a\right)=0& \cr \color{green}{\Leftrightarrow}&\left(x+a-a\right)\cdot \left(x+a+a\right)=0& \cr \color{green}{\Leftrightarrow}&{\left(x+a\right)}^2-a^2=0& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[x^3-1=0,(x-1)*(x^2+x+1)=0,x=1
]
[]
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
[x^3-1=0,(x-1)*(x^2+x+1)=0,x=1
or x^2+x+1=0,x=1 or x = -(sqr
t(3)*%i+1)/2 or x=(sqrt(3)*%i-
1)/2]
[]
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
[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 $\begin{array}{lll} &a\cdot x^2+b\cdot x+c=0\,{\mbox{ or }}\, a=0& \cr \color{green}{\Leftrightarrow}&a^2\cdot x^2+a\cdot b\cdot x+a\cdot c=0& \cr \color{green}{\Leftrightarrow}&{\left(a\cdot x\right)}^2+b\cdot \left(a\cdot x\right)+a\cdot c=0& \cr \color{green}{\Leftrightarrow}&{\left(a\cdot x\right)}^2+b\cdot \left(a\cdot x\right)+\frac{b^2}{4}-\frac{b^2}{4}+a\cdot c=0& \cr \color{green}{\Leftrightarrow}&{\left(a\cdot x+\frac{b}{2}\right)}^2-\frac{b^2}{4}+a\cdot c=0& \cr \color{green}{\Leftrightarrow}&{\left(a\cdot x+\frac{b}{2}\right)}^2=\frac{b^2}{4}-a\cdot c& \cr \color{green}{\Leftrightarrow}&a\cdot x+\frac{b}{2}= \pm \sqrt{\frac{b^2}{4}-a\cdot c}& \cr \color{green}{\Leftrightarrow}&a\cdot x=-\frac{b}{2}+\sqrt{\frac{b^2}{4}-a\cdot c}\,{\mbox{ or }}\, a\cdot x=-\frac{b}{2}-\sqrt{\frac{b^2}{4}-a\cdot c}& \cr \color{green}{\Leftrightarrow}&a=0\,{\mbox{ or }}\, x=\frac{-b+\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}\,{\mbox{ or }}\, \left(a=0\,{\mbox{ or }}\, x=\frac{-b-\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}\right)& \cr \color{green}{\Leftrightarrow}&a^2=0\,{\mbox{ or }}\, x=\frac{-b+\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}\,{\mbox{ or }}\, x=\frac{-b-\sqrt{b^2-4\cdot a\cdot c}}{2\cdot a}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &a\cdot x^2+b\cdot x=-c& \cr \color{red}{\Rightarrow}&4\cdot a^2\cdot x^2+4\cdot a\cdot b\cdot x+b^2=b^2-4\cdot a\cdot c& \cr \color{green}{\Leftrightarrow}&{\left(2\cdot a\cdot x+b\right)}^2=b^2-4\cdot a\cdot c& \cr \color{green}{\Leftrightarrow}&2\cdot a\cdot x+b= \pm \sqrt{b^2-4\cdot a\cdot c}& \cr \color{green}{\Leftrightarrow}&2\cdot a\cdot x={-b \pm \sqrt{b^2-4\cdot a\cdot c}}& \cr \color{red}{?}&x=\frac{{-b \pm \sqrt{b^2-4\cdot a\cdot c}}}{2\cdot a}& \cr \end{array}$ (EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR)
Equiv
[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#sqr
t(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 $\begin{array}{lll} &a\cdot x^2+b\cdot x=-c\,{\mbox{ or }}\, a=0& \cr \color{green}{\Leftrightarrow}&4\cdot a^2\cdot x^2+4\cdot a\cdot b\cdot x+b^2=b^2-4\cdot a\cdot c& \cr \color{green}{\Leftrightarrow}&{\left(2\cdot a\cdot x+b\right)}^2=b^2-4\cdot a\cdot c& \cr \color{green}{\Leftrightarrow}&2\cdot a\cdot x+b= \pm \sqrt{b^2-4\cdot a\cdot c}& \cr \color{green}{\Leftrightarrow}&2\cdot a\cdot x={-b \pm \sqrt{b^2-4\cdot a\cdot c}}& \cr \color{green}{\Leftrightarrow}&x=\frac{{-b \pm \sqrt{b^2-4\cdot a\cdot c}}}{2\cdot a}\,{\mbox{ or }}\, a=0& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &\sqrt{3\cdot x+4}=2+\sqrt{x+2}&{\color{blue}{{x \in {\left[ -\frac{4}{3},\, \infty \right)}}}}\cr \color{red}{\Rightarrow}&3\cdot x+4=4+4\cdot \sqrt{x+2}+\left(x+2\right)&{\color{blue}{{x \in {\left[ -2,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&x-1=2\cdot \sqrt{x+2}&{\color{blue}{{x \in {\left[ -2,\, \infty \right)}}}}\cr \color{red}{\Rightarrow}&x^2-2\cdot x+1=4\cdot x+8& \cr \color{green}{\Leftrightarrow}&x^2-6\cdot x-7=0& \cr \color{green}{\Leftrightarrow}&\left(x-7\right)\cdot \left(x+1\right)=0& \cr \color{green}{\Leftrightarrow}&x=7\,{\mbox{ or }}\, x=-1& \cr \end{array}$ (EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&\sqrt{3\cdot x+4}=2+\sqrt{x+2}&{\color{blue}{{x \in {\left[ 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&3\cdot x+4=4+4\cdot \sqrt{x+2}+\left(x+2\right)&{\color{blue}{{x \in {\left[ 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&x-1=2\cdot \sqrt{x+2}&{\color{blue}{{x \in {\left[ 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&x^2-2\cdot x+1=4\cdot x+8& \cr \color{green}{\Leftrightarrow}&x^2-6\cdot x-7=0& \cr \color{green}{\Leftrightarrow}&\left(x-7\right)\cdot \left(x+1\right)=0& \cr \color{green}{\Leftrightarrow}&x=7\,{\mbox{ or }}\, x=-1& \cr \color{green}{\Leftrightarrow}&x=7& \cr \end{array}$ (ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[x*(x-1)*(x-2)=0,x*(x-1)=0,x*(
x-1)*(x-2)=0,x*(x^2-2)=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
[x^2-6*x=-9,x=3]
[]
1 $\begin{array}{lll} &x^2-6\cdot x=-9& \cr \color{green}{\mbox{(Same roots)}}&x=3& \cr \end{array}$ (EMPTYCHAR,SAMEROOTS)
Equiv
[x=1 nounor x=-2 nounor x=1,x^
3-3*x=-2,x=1 nounor x=-2]
[]
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
[9*x^3-24*x^2+13*x=2,x=1/3 nou
nor x=2]
[]
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
[(x-2)^43*(x+1/3)^60=0,(3*x+1)
^4*(x-2)^2=0,x=-1/3 nounor x=2
]
[]
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
[2^x=4,x*log(2)=log(4),x=log(2
^2)/log(2),x=2*log(2)/log(2),x
=2]
[]
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
[x^log(y),stackeq(e^(log(x)*lo
g(y))),stackeq(e^(log(y)*log(x
))),stackeq(y^log(x))]
[]
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
[lg(x+17,3)-2=lg(2*x,3),lg(x+1
7,3)-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 $\begin{array}{lll} &\log_{3}\left(x+17\right)-2=\log_{3}\left(2\cdot x\right)&{\color{blue}{{x \in {\left( 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&\log_{3}\left(x+17\right)-\log_{3}\left(2\cdot x\right)=2&{\color{blue}{{x \in {\left( 0,\, \infty \right)}}}}\cr \color{green}{\Leftrightarrow}&\log_{3}\left(\frac{x+17}{2\cdot x}\right)=2& \cr \color{green}{\log(?)}&\frac{x+17}{2\cdot x}=3^2&{\color{blue}{{x \not\in {\left \{0 \right \}}}}}\cr \color{green}{\Leftrightarrow}&x+17=18\cdot x& \cr \color{green}{\Leftrightarrow}&17\cdot x=17& \cr \color{green}{\Leftrightarrow}&x=1& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,EQUIVLOG, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[a=logbase(9,3),3^a=9,3^a=3^2,
a=2]
[]
1 $\begin{array}{lll} &a=\log_{3}\left(9\right)& \cr \color{green}{\Leftrightarrow}&3^{a}=9& \cr \color{green}{\Leftrightarrow}&3^{a}=3^2& \cr \color{green}{\Leftrightarrow}&a=2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[x=(1+y/n)^n,x^(1/n)=(1+y/n),y
/n=x^(1/n)-1,y=n*(x^(1/n)-1)]
[]
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
[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 $\begin{array}{lll}\color{blue}{(\mathbb{R})}&a^3=b^3& \cr \color{green}{\Leftrightarrow}&a^3-b^3=0& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a^2+a\cdot b+b^2\right)=0& \cr \color{red}{\Leftarrow}&a-b=0& \cr \color{green}{\Leftrightarrow}&a=b& \cr \end{array}$ (ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR)
Equiv
[x^3-1=0,(x-1)*(x^2+x+1)=0,x=1
]
[]
[assumereal]
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
[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 $\begin{array}{lll}\color{blue}{(\mathbb{R})}&x^4=2& \cr \color{green}{\Leftrightarrow}&x^4-2=0& \cr \color{green}{\Leftrightarrow}&\left(x^2-\sqrt{2}\right)\cdot \left(x^2+\sqrt{2}\right)=0& \cr \color{green}{\Leftrightarrow}\, \color{blue}{(\mathbb{R})}&x^2=\sqrt{2}& \cr \color{green}{\Leftrightarrow}&x= \pm 2^{\frac{1}{4}}& \cr \end{array}$ (ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHARREAL, EQUIVCHAR)
Equiv
[6*x-12=3*(x-2),6*x-12+3*(x-2)
=0,9*x-18=0,x=2]
[]
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
[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 $\begin{array}{lll} &x^2-6\cdot x+9=0& \cr \color{green}{\Leftrightarrow}&x^2-6\cdot x=-9& \cr \color{green}{\Leftrightarrow}&x\cdot \left(x-6\right)=3\cdot \left(-3\right)& \cr \color{green}{\Leftrightarrow}&x=3\,{\mbox{ or }}\, x-6=-3& \cr \color{green}{\mbox{(Same roots)}}&x=3& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,SAMEROOTS)
Equiv
[(x+3)*(2-x)=4,x+3=4 or (2-x)=
4,x=1 or x=-2]
[]
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
[(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 $\begin{array}{lll} &\left(x-p\right)\cdot \left(x-q\right)=0& \cr \color{green}{\Leftrightarrow}&x^2-p\cdot x+\left(-q\right)\cdot x+p\cdot q=0& \cr \color{green}{\Leftrightarrow}&1+q-x-p+\left(-p\right)\cdot q+p\cdot x+x+q\cdot x-x^2=1-p+q& \cr \color{green}{\Leftrightarrow}&\left(1+q-x\right)\cdot \left(1-p+x\right)=1-p+q& \cr \color{green}{\Leftrightarrow}&1+q-x=1-p+q\,{\mbox{ or }}\, 1-p+x=1-p+q& \cr \color{green}{\Leftrightarrow}&x=p\,{\mbox{ or }}\, x=q& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &a=b& \cr \color{red}{\Rightarrow}&a^2=a\cdot b& \cr \color{green}{\Leftrightarrow}&a^2-b^2=a\cdot b-b^2& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a+b\right)=b\cdot \left(a-b\right)& \cr \color{red}{\Leftarrow}&a+b=b& \cr \color{green}{\Leftrightarrow}&2\cdot a=a& \cr \color{red}{\Leftarrow}&1=2& \cr \end{array}$ (EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR,IMPLIEDCHAR)
Equiv
[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 $\begin{array}{lll} &a=b\,{\mbox{ or }}\, a=0& \cr \color{green}{\Leftrightarrow}&a^2=a\cdot b& \cr \color{green}{\Leftrightarrow}&a^2-b^2=a\cdot b-b^2& \cr \color{green}{\Leftrightarrow}&\left(a-b\right)\cdot \left(a+b\right)=b\cdot \left(a-b\right)& \cr \color{green}{\Leftrightarrow}&a+b=b\,{\mbox{ or }}\, a-b=0& \cr \color{green}{\Leftrightarrow}&2\cdot a=a\,{\mbox{ or }}\, a=b& \cr \color{green}{\Leftrightarrow}&2=1\,{\mbox{ or }}\, a=0\,{\mbox{ or }}\, a=b& \cr \color{green}{\Leftrightarrow}&a=0\,{\mbox{ or }}\, a=b& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[(x^2-4)/(x-2)=0,(x-2)*(x+2)/(
x-2)=0,x+2=0,x=-2]
[]
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
[(x^2-4)/(x-2)=0,(x^2-4)=0,(x-
2)*(x+2)=0,x=-2 or x=2]
[]
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
[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 $\begin{array}{lll} &\frac{5\cdot x}{2\cdot x+1}-\frac{3}{x+1}=1&{\color{blue}{{x \not\in {\left \{-1 , -\frac{1}{2} \right \}}}}}\cr \color{green}{\Leftrightarrow}&5\cdot x\cdot \left(x+1\right)-3\cdot \left(2\cdot x+1\right)=\left(x+1\right)\cdot \left(2\cdot x+1\right)& \cr \color{green}{\Leftrightarrow}&5\cdot x^2+5\cdot x-6\cdot x-3=2\cdot x^2+3\cdot x+1& \cr \color{green}{\Leftrightarrow}&3\cdot x^2-4\cdot x-4=0& \cr \color{green}{\Leftrightarrow}&\left(x-2\right)\cdot \left(3\cdot x+2\right)=0& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=\frac{-2}{3}& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[(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 $\begin{array}{lll} &\frac{x+10}{x-6}-5=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{6 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{x+10-5\cdot \left(x-6\right)}{x-6}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{6 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{4\cdot x-40}{6-x}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{6 , 13 \right \}}}}}\cr \color{red}{?}&6-x=13-x& \cr \color{green}{\Leftrightarrow}&6=13& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR, EQUIVCHAR)
Equiv
[(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 $\begin{array}{lll} &\frac{x+5}{x-7}-5=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{x+5-5\cdot \left(x-7\right)}{x-7}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{green}{\Leftrightarrow}&\frac{4\cdot x-40}{7-x}=\frac{4\cdot x-40}{13-x}&{\color{blue}{{x \not\in {\left \{7 , 13 \right \}}}}}\cr \color{red}{\Leftarrow}&7-x=13-x& \cr \color{green}{\Leftrightarrow}&7=13& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR)
Equiv
[(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-4
0=0,7= 13 or 4*x=40,x=10]
[]
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
[a*x^2+b*x+c=0,a=0 nounand b=0
nounand c=0,a*x^2+b*x+c=0]
[]
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
[a*x^2+b*x+c=A*x^2+B*x+C,a=A n
ounand b=B nounand c=C,a*x^2+b
*x+c=A*x^2+B*x+C]
[]
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
[(x-1)*(x+4), stackeq(x^2-x+4*
x-4),stackeq(x^2+3*x-4)]
[]
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
[(x-1)*(x+4), stackeq(x^2-x+4*
x-4),stackeq(x^2+3*x-4)]
[]
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
[x^2-2,stackeq((x-sqrt(2))*(x+
sqrt(2)))]
[]
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
[x^2+4,stackeq((x-2*i)*(x+2*i)
)]
[]
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
[x^2+2*a*x,x^2+2*a*x+a^2-a^2,(
x+a)^2-a^2]
[]
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
[x^2+2*a*x,stackeq(x^2+2*a*x+a
^2-a^2),stackeq((x+a)^2-a^2)]
[]
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
[(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 $\begin{array}{lll} &\frac{y-z}{y\cdot z}+\frac{z-x}{z\cdot x}+\frac{x-y}{x\cdot y}& \cr \color{green}{\Leftrightarrow}&\frac{x\cdot \left(y-z\right)+y\cdot \left(z-x\right)+z\cdot \left(x-y\right)}{x\cdot y\cdot z}& \cr \color{green}{\Leftrightarrow}&0& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[(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 $\begin{array}{lll} &\frac{y-z}{y\cdot z}+\frac{z-x}{z\cdot x}+\frac{x-y}{x\cdot y}& \cr \color{green}{\checkmark}&=\frac{x\cdot \left(y-z\right)+y\cdot \left(z-x\right)+z\cdot \left(x-y\right)}{x\cdot y\cdot z}& \cr \color{green}{\checkmark}&=0& \cr \end{array}$ (EMPTYCHAR, CHECKMARK, CHECKMARK)
Equiv
[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 $\begin{array}{lll} &2\cdot \left(a^2\cdot b^2+b^2\cdot c^2+c^2\cdot a^2\right)-\left(a^4+b^4+c^4\right)& \cr \color{green}{\checkmark}&=4\cdot a^2\cdot b^2-\left(a^4+b^4+c^4+2\cdot a^2\cdot b^2-2\cdot b^2\cdot c^2-2\cdot c^2\cdot a^2\right)& \cr \color{green}{\checkmark}&={\left(2\cdot a\cdot b\right)}^2-{\left(b^2+a^2-c^2\right)}^2& \cr \color{green}{\checkmark}&=\left({\left(a+b\right)}^2-c^2\right)\cdot \left(c^2-{\left(a-b\right)}^2\right)& \cr \color{green}{\checkmark}&=\left(a+b+c\right)\cdot \left(a+b-c\right)\cdot \left(c+a-b\right)\cdot \left(c-a+b\right)& \cr \end{array}$ (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK)
Equiv
[abs(x-1/2)+abs(x+1/2)-2,stack
eq(abs(x)-1)]
[]
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
[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 $\begin{array}{lll} &11\cdot \sqrt{\left| x\right| +1}=25-x& \cr \color{red}{?}&11^2\cdot \left(\left| x\right| +1\right)={\left(25-x\right)}^2& \cr \color{green}{\Leftrightarrow}&11^2\cdot \left| x\right| ={\left(25-x\right)}^2-11^2& \cr \color{green}{\Leftrightarrow}&11^4\cdot x^2={\left({\left(25-x\right)}^2-11^2\right)}^2& \cr \color{green}{\Leftrightarrow}&{\left({\left(25-x\right)}^2-11^2\right)}^2-11^4\cdot x^2=0& \cr \color{green}{\Leftrightarrow}&\left({\left(25-x\right)}^2-11^2+\left(-11^2\right)\cdot x\right)\cdot \left({\left(25-x\right)}^2-11^2+11^2\cdot x\right)=0& \cr \color{green}{\Leftrightarrow}&\left(x^2-50\cdot x+504-121\cdot x\right)\cdot \left(x^2-50\cdot x+504+121\cdot x\right)=0& \cr \color{green}{\Leftrightarrow}&\left(x-168\right)\cdot \left(x-3\right)\cdot \left(x+8\right)\cdot \left(x+63\right)=0& \cr \end{array}$ (EMPTYCHAR,QMCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[1/(x^2+1)=1/((x+%i)*(x-%i)),
stackeq(1/(2*%i)*(1/(x-%i)-1/(
x+%i)))]
[]
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
[((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 $\begin{array}{lll} &\frac{\frac{a-b}{a^2+a\cdot b}}{\frac{a^2-2\cdot a\cdot b+b^2}{a^4-b^4}}& \cr \color{green}{\checkmark}&=\frac{\left(a-b\right)\cdot \left(a-b\right)\cdot \left(a+b\right)\cdot \left(a^2+b^2\right)}{a\cdot \left(a+b\right)\cdot {\left(a-b\right)}^2}& \cr \color{green}{\checkmark}&=\frac{a^2+b^2}{a}& \cr \color{green}{\checkmark}&=a+\frac{b^2}{a}& \cr \end{array}$ (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK)
Equiv
[a^4+4*b^4,stackeq((a^2)^2+4*a
^2*b^2+(2*b^2)^2-4*a^2*b^2),st
ackeq((a^2+2*b^2)^2-(2*a*b)^2)
,stackeq((2*b^2-2*a*b+a^2)*(2*
b^2+2*a*b+a^2))]
[]
1 $\begin{array}{lll} &a^4+4\cdot b^4& \cr \color{green}{\checkmark}&={\left(a^2\right)}^2+4\cdot a^2\cdot b^2+{\left(2\cdot b^2\right)}^2-4\cdot a^2\cdot b^2& \cr \color{green}{\checkmark}&={\left(a^2+2\cdot b^2\right)}^2-{\left(2\cdot a\cdot b\right)}^2& \cr \color{green}{\checkmark}&=\left(2\cdot b^2-2\cdot a\cdot b+a^2\right)\cdot \left(2\cdot b^2+2\cdot a\cdot b+a^2\right)& \cr \end{array}$ (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK)
Equiv
[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 $\begin{array}{lll} &\sum_{k=1}^{n+1}{k}& \cr \color{green}{\checkmark}&=\sum_{k=1}^{n}{k}+\left(n+1\right)& \cr \color{green}{\checkmark}&=\frac{n\cdot \left(n+1\right)}{2}+n+1& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)\cdot \left(n+1+1\right)}{2}& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)\cdot \left(n+2\right)}{2}& \cr \end{array}$ (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK)
Equiv
[log((a-1)^n*product(x_i^(-a),
i,1,n)),stackeq(n*log(a-1)-a*s
um(log(x_i),i,1,n))]
[]
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
[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 $\begin{array}{lll} &{{n}\choose{k}}+{{n}\choose{k+1}}& \cr \color{green}{\checkmark}&=\frac{n!}{k!\cdot \left(n-k\right)!}+\frac{n!}{\left(k+1\right)!\cdot \left(n-k-1\right)!}& \cr \color{green}{\checkmark}&=\frac{n!}{k!\cdot \left(n-k\right)\cdot \left(n-k-1\right)!}+\frac{n!}{\left(k+1\right)!\cdot \left(n-k-1\right)!}& \cr \color{green}{\checkmark}&=\frac{n!}{k!\cdot \left(n-k-1\right)!}\cdot \left(\frac{1}{n-k}+\frac{1}{k+1}\right)& \cr \color{green}{\checkmark}&=\frac{n!}{k!\cdot \left(n-k-1\right)!}\cdot \left(\frac{n+1}{\left(n-k\right)\cdot \left(k+1\right)}\right)& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)\cdot n!}{k!\cdot \left(n-k-1\right)!}\cdot \left(\frac{1}{\left(k+1\right)\cdot \left(n-k\right)}\right)& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)\cdot n!}{\left(k+1\right)\cdot k!\cdot \left(n-k\right)\cdot \left(n-k-1\right)!}& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)!}{\left(k+1\right)!}\cdot \left(\frac{1}{\left(n-k\right)\cdot \left(n-k-1\right)!}\right)& \cr \color{green}{\checkmark}&=\frac{\left(n+1\right)!}{\left(k+1\right)!\cdot \left(n-k\right)!}& \cr \color{green}{\checkmark}&={{n+1}\choose{k+1}}& \cr \end{array}$ (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK)
Equiv
[(x-1)^2=(x-1)*(x-1), stackeq(
x^2-2*x+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
[(x-1)^2=(x-1)*(x-1), stackeq(
x^2-2*x+2)]
[]
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
[(x-2)^2=(x-1)*(x-1), stackeq(
x^2-2*x+1)]
[]
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
[4^((n+1)+1)-1= 4*4^(n+1)-1,st
ackeq(4*(4^(n+1)-1)+3)]
[]
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
[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 $\begin{array}{lll} &\left\{\begin{array}{l}2\cdot x+3\cdot y=6\cr 4\cdot x+9\cdot y=15\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}2\cdot x+3\cdot y=6\cr -2\cdot x=-3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}3+3\cdot y=6\cr 2\cdot x=3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}y=1\cr x=\frac{3}{2}\cr \end{array}\right.& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &\left\{\begin{array}{l}2\cdot x+3\cdot y=6\cr 4\cdot x+9\cdot y=15\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}2\cdot x+3\cdot y=6\cr -2\cdot x=-3\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}3+3\cdot y=6\cr 2\cdot x=3\cr \end{array}\right.& \cr \color{red}{?}&\left\{\begin{array}{l}y=1\cr x=3\cr \end{array}\right.& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR)
Equiv
[x^2+y^2=8 and x=y, 2*x^2=8 an
d 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 $\begin{array}{lll} &\left\{\begin{array}{l}x^2+y^2=8\cr x=y\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}2\cdot x^2=8\cr y=x\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x^2=4\cr y=x\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x= \pm 2\cr y=x\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ and }}\, y=x\,{\mbox{ or }}\, x=-2\,{\mbox{ and }}\, y=x& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ and }}\, y=2\,{\mbox{ or }}\, x=-2\,{\mbox{ and }}\, y=-2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 an
d 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 $\begin{array}{lll} &\left\{\begin{array}{l}x^2+y^2=5\cr x\cdot y=2\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x^2+y^2-5=0\cr x\cdot y-2=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x^2-2\cdot x\cdot y+y^2-1=0\cr x^2+2\cdot x\cdot y+y^2-9=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}{\left(x-y\right)}^2-1=0\cr {\left(x+y\right)}^2-3^2=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&x-y=1\,{\mbox{ and }}\, x+y=3\,{\mbox{ or }}\, x-y=-1\,{\mbox{ and }}\, x+y=3\,{\mbox{ or }}\, x-y=1\,{\mbox{ and }}\, x+y=-3\,{\mbox{ or }}\, x-y=-1\,{\mbox{ and }}\, x+y=-3& \cr \color{green}{\Leftrightarrow}&x=1\,{\mbox{ and }}\, y=2\,{\mbox{ or }}\, x=2\,{\mbox{ and }}\, y=1\,{\mbox{ or }}\, x=-2\,{\mbox{ and }}\, y=-1\,{\mbox{ or }}\, x=-1\,{\mbox{ and }}\, y=-2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &\left\{\begin{array}{l}4\cdot x^2+7\cdot x\cdot y+4\cdot y^2=4\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}4\cdot x^2+7\cdot x\cdot \left(x-4\right)+4\cdot {\left(x-4\right)}^2-4=0\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}15\cdot x^2-60\cdot x+60=0\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}{\left(x-2\right)}^2=0\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x=2\cr y=x-4\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}x=2\cr y=-2\cr \end{array}\right.& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &\left\{\begin{array}{l}a^2=b\cr a^2=1\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a=1\,{\mbox{ or }}\, a=-1\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&b=1\,{\mbox{ and }}\, a=1\,{\mbox{ or }}\, b=1\,{\mbox{ and }}\, a=-1& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[a^2=b and x=1, b=a^2 and x=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
[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 $\begin{array}{lll}\color{blue}{(\mathbb{R})}&\left\{\begin{array}{l}a^2=b\cr b^2=a\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a^4=a\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a^4-a=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a\cdot \left(a-1\right)\cdot \left(a^2+a+1\right)=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&\left\{\begin{array}{l}b=a^2\cr a=0\,{\mbox{ or }}\, a=1\,{\mbox{ or }}\, a^2+a+1=0\cr \end{array}\right.& \cr \color{green}{\Leftrightarrow}&b=0\,{\mbox{ and }}\, a=0\,{\mbox{ or }}\, b=1\,{\mbox{ and }}\, a=1& \cr \end{array}$ (ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &2\cdot x^3-9\cdot x^2+10\cdot x-3& \cr &\mbox{Let }x = 1& \cr \color{green}{\Leftrightarrow}&2\cdot 1^3-9\cdot 1^2+10\cdot 1-3& \cr \color{green}{\checkmark}&=0& \cr &\mbox{So}& \cr &2\cdot x^3-9\cdot x^2+10\cdot x-3& \cr \color{green}{\checkmark}&=\left(x-1\right)\cdot \left(2\cdot x^2-7\cdot x+3\right)& \cr \color{green}{\checkmark}&=\left(x-1\right)\cdot \left(2\cdot x-1\right)\cdot \left(x-3\right)& \cr \end{array}$ (EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, CHECKMARK, EMPTYCHAR, EMPTYCHAR, CHECKMARK, CHECKMARK)
Equiv
[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 $\begin{array}{lll} &2\cdot x^2+x\geq 6& \cr \color{green}{\Leftrightarrow}&2\cdot x^2+x-6\geq 0& \cr \color{green}{\Leftrightarrow}&\left(2\cdot x-3\right)\cdot \left(x+2\right)\geq 0& \cr \color{green}{\Leftrightarrow}&2\cdot x-3\geq 0\,{\mbox{ and }}\, x+2\geq 0\,{\mbox{ or }}\, 2\cdot x-3\leq 0\,{\mbox{ and }}\, x+2\leq 0& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ and }}\, x\geq -2\,{\mbox{ or }}\, x\leq \frac{3}{2}\,{\mbox{ and }}\, x\leq -2& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ or }}\, x\leq -2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &2\cdot x^2+x\geq 6& \cr \color{green}{\Leftrightarrow}&2\cdot x^2+x-6\geq 0& \cr \color{green}{\Leftrightarrow}&\left(2\cdot x-3\right)\cdot \left(x+2\right)\geq 0& \cr \color{green}{\Leftrightarrow}&2\cdot x-3\geq 0\,{\mbox{ and }}\, x+2\geq 0\,{\mbox{ or }}\, 2\cdot x-3\leq 0\,{\mbox{ and }}\, x+2\leq 0& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ and }}\, x\geq -2\,{\mbox{ or }}\, x\leq \frac{3}{2}\,{\mbox{ and }}\, x\leq -2& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ or }}\, x\leq -2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[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 $\begin{array}{lll} &2\cdot x^2+x\geq 6& \cr \color{green}{\Leftrightarrow}&2\cdot x^2+x-6\geq 0& \cr \color{green}{\Leftrightarrow}&\left(2\cdot x-3\right)\cdot \left(x+2\right)\geq 0& \cr \color{green}{\Leftrightarrow}&2\cdot x-3\geq 0\,{\mbox{ and }}\, x+2\geq 0\,{\mbox{ or }}\, 2\cdot x-3\leq 0\,{\mbox{ and }}\, x+2\leq 0& \cr \color{green}{\Leftrightarrow}&x\geq \frac{3}{2}\,{\mbox{ and }}\, x\geq -2\,{\mbox{ or }}\, x\leq \frac{3}{2}\,{\mbox{ and }}\, x\leq -2& \cr \color{red}{?}&x\geq \frac{3}{2}\,{\mbox{ or }}\, x\leq 2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR)
Equiv
[x^2>=9 and x>3, x^2-9&g
t;=0 and x>3, (x>=3 or x
<=-3) and x>3, x>3]
[]
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
[-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 t
rue for all x.", "So
it must be true when the righ
t hand side takes its minimum
value.", "This happe
ns 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&l
t;a and a<2) or false, (-6&
lt;a and a<2)]
[]
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
[x-2>0 and x*(x-2)<15,x&
gt;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 an
d (x<5 and x>-3),x>2
and x<5]
[]
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
[x-2>0 and x*(x-2)<15,x&
gt;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 an
d (x<5 and x>-3),x>2
and x<5]
[]
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
[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&
gt;=12,a-4>=sqrt(12) or a-4
<= -sqrt(12),"Ignoring
the negative solution.",
a>=sqrt(12)+4,"Using e
xternal domain information tha
t a is an integer.",a>
=8]
[]
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
[x^2#1,x^2-1#0,(x-1)*(x+1)#0,x
<-1 nounor (-1<x nounand
x<1) nounor x>1]
[]
1 $\begin{array}{lll} &x^2\neq 1& \cr \color{green}{\Leftrightarrow}&x^2-1\neq 0& \cr \color{green}{\Leftrightarrow}&\left(x-1\right)\cdot \left(x+1\right)\neq 0& \cr \color{green}{\Leftrightarrow}&x < -1\,{\mbox{ or }}\, -1 < x\,{\mbox{ and }}\, x < 1\,{\mbox{ or }}\, x > 1& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
["Set P(n) be the stateme
nt 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 as
sume 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 $\begin{array}{lll} &\mbox{Set P(n) be the statement that}& \cr &\sum_{k=1}^{n}{k^2}=\frac{n\cdot \left(n+1\right)\cdot \left(2\cdot n+1\right)}{6}& \cr &\mbox{Then P(1) is the statement}& \cr &1^2=\frac{1\cdot \left(1+1\right)\cdot \left(2\cdot 1+1\right)}{6}& \cr \color{green}{\Leftrightarrow}&1=1& \cr &\mbox{So P(1) holds. Now assume P(n) is true.}& \cr &\sum_{k=1}^{n}{k^2}=\frac{n\cdot \left(n+1\right)\cdot \left(2\cdot n+1\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n}{k^2}+{\left(n+1\right)}^2=\frac{n\cdot \left(n+1\right)\cdot \left(2\cdot n+1\right)}{6}+{\left(n+1\right)}^2& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(n\cdot \left(2\cdot n+1\right)+6\cdot \left(n+1\right)\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(2\cdot n^2+7\cdot n+6\right)}{6}& \cr \color{green}{\Leftrightarrow}&\sum_{k=1}^{n+1}{k^2}=\frac{\left(n+1\right)\cdot \left(n+1+1\right)\cdot \left(2\cdot \left(n+1\right)+1\right)}{6}& \cr \end{array}$ (EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
Equiv
[(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, s
um(k^2,k,1,n+1) = ((n+1)*(n+2)
*(2*(n+1)+1))/6]
[]
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
[conjugate(a)*conjugate(b),sta
cklet(a,x+i*y),stacklet(b,r+i*
s),stackeq(conjugate(x+i*y)*co
njugate(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))),stacke
q(conjugate((x+i*y)*(r+i*s))),
stacklet(x+i*y,a),stacklet(r+i
*s,b),stackeq(conjugate(a*b))]
[]
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
[nounint(x*e^x,x,-inf,0),nounl
imit(nounint(x*e^x,x,t,0),t,-i
nf),nounlimit(e^t-t*e^t-1,t,-i
nf),nounlimit(e^t,t,-inf)+noun
limit(-t*e^t,t,-inf)+nounlimit
(-1,t,-inf),-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
[noundiff(x^2,x),stackeq(nounl
imit(((x+h)^2-x^2)/h,h,0)),sta
ckeq(nounlimit(2*x+h,h,0)),sta
ckeq(2*x)]
[]
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
[-12+3*noundiff(y(x),x)+8-8*no
undiff(y(x),x)=0,-5*noundiff(y
(x),x)=4,noundiff(y(x),x)=-4/5
]
[]
[calculus]
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
[x^2+1,x^3/3+x,x^2+1,x^3/3+x+c
]
[]
[calculus]
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
[3*x^(3/2)-2/x,(9*sqrt(x))/2+2
/x^2,3*x^(3/2)-2/x+c]
[]
[calculus]
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
[x^2+1,stackeq(x^3/3+x),stacke
q(x^2+1),stackeq(x^3/3+x+c)]
[]
[calculus]
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
[diff(x^2*sin(x),x),stackeq(x^
2*diff(sin(x),x)+diff(x^2,x)*s
in(x)),stackeq(x^2*cos(x)+2*x*
sin(x))]
[]
[calculus]
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
[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)*si
n(x)+6,diff(y(x),x) = (y(x)*si
n(x)+6)/(cos(x)+2*y(x))]
[]
[calculus]
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
[nounint(s^2+1,s),stackeq(s^3/
3+s+c)]
[]
[calculus]
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
[nounint(x^3*log(x),x),x^4/4*l
og(x)-1/4*nounint(x^3,x),x^4/4
*log(x)-x^4/16]
[]
[calculus]
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
[nounint(x^3*log(x),x),x^4/4*l
og(x)-1/4*nounint(x^3,x),x^4/4
*log(x)-x^4/16+c]
[]
[calculus]
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
[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 $\begin{array}{lll} &\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{2}{x}\cdot y=x^3\cdot \sin \left( 3\cdot x \right)& \cr \color{green}{\Leftrightarrow}&\frac{1}{x^2}\cdot \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)-\frac{2}{x^3}\cdot y=x\cdot \sin \left( 3\cdot x \right)& \cr \color{green}{\Leftrightarrow}&\left(\frac{\mathrm{d}}{\mathrm{d} x} \frac{y}{x^2}\right)=x\cdot \sin \left( 3\cdot x \right)& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{y}{x^2}=\int {x\cdot \sin \left( 3\cdot x \right)}{\;\mathrm{d}x}& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{y}{x^2}=\frac{\sin \left( 3\cdot x \right)-3\cdot x\cdot \cos \left( 3\cdot x \right)}{9}+c& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,INTCHAR(x),INTCHAR(x))

## EquivFirst

Test
?
Student response
Opt
Mark
CAS errors
Feedback
EquivFirst
x
[x^2=4,x=2 or x=-2]
EquivFirst
[x^2=4,x=2 or x=-2]
x
EquivFirst
[1/0]
[x^2=4,x=2 or x=-2]
-1 ATEquivFirst_STACKERROR_SAns.
EquivFirst
[x^2=4,x=2 or x=-2]
[1/0]
-1 ATEquivFirst_STACKERROR_TAns.
EquivFirst
[x^2=4,x=2 or x=-2]
[x^2=4,x=2 or x=-2]
1 $\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR)
EquivFirst
[x^2=9,x=3 or x=-3]
[x^2=4,x=2 or x=-2]
0 The first line in your argument must be "$$x^2=4$$". ATEquivFirst_SA_wrong_start
EquivFirst
[x^2=4,x=2]
[x^2=4,x=2 or x=-2]
0 $\begin{array}{lll} &x^2=4& \cr \color{red}{\Leftarrow}&x=2& \cr \end{array}$ (EMPTYCHAR,IMPLIEDCHAR)
EquivFirst
[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 $\begin{array}{lll} &x^2=4& \cr \color{green}{\Leftrightarrow}&x^2-4=0& \cr \color{green}{\Leftrightarrow}&\left(x-2\right)\cdot \left(x+2\right)=0& \cr \color{green}{\Leftrightarrow}&x=2\,{\mbox{ or }}\, x=-2& \cr \end{array}$ (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
EquivFirst
[x^2=4,x= #pm#2, x=2 or x=-2]
[x^2=4,x=2 or x=-2]
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
[x^2-6*x+9=0,x=3]
[x^2-6*x+9=0,x=3]
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
[x^2=4,x=2]
[x^2=4,x=2]
[assumepos]
1 $\begin{array}{lll}\color{blue}{\mbox{Assume +ve vars}}&x^2=4& \cr \color{green}{\Leftrightarrow}&x=2& \cr \end{array}$ (ASSUMEPOSVARS, EQUIVCHAR)

## SingleFrac

Test
?
Student response
Opt
Mark
CAS errors
Feedback
SingleFrac
1/0
1/n
-1 ATSingleFrac_STACKERROR_SAns.
SingleFrac
0
1/0
-1 ATSingleFrac_STACKERROR_TAns.
SingleFrac
x=3
2
0 Your answer should be an expression, not an equation, inequality, list, set or matrix. ATSingleFrac_SA_not_expression.
SingleFrac
3
3
1
SingleFrac
3
2
0 Your answer is not algebraically equivalent to the correct answer. You must have done something wrong. ATSingleFrac_ret_exp.
SingleFrac
1/m
1/n
0 Your answer is not algebraically equivalent to the correct answer. You must have done something wrong. ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac
1/n
1/n
1 ATSingleFrac_true.
SingleFrac
a+1/2
(2*a+1)/2
0 Your answer needs to be a single fraction of the form $${a}\over{b}$$. ATSingleFrac_part.
SingleFrac
a+1/2
(2*a+1)/2
0 Your answer needs to be a single fraction of the form $${a}\over{b}$$. ATSingleFrac_part.
SingleFrac
4/(x^2+2*x-24)+2/(x^2+4*x-12)
(6*x-16)/(x^3-28*x+48)
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
2*(1/n)
2/n
0 Your answer needs to be a single fraction of the form $${a}\over{b}$$. ATSingleFrac_part.
SingleFrac
2/n
2/n
1 ATSingleFrac_true.
Simple Mistakes
SingleFrac
2/(n+1)
1/(n+1)
0 Your answer is not algebraically equivalent to the correct answer. You must have done something wrong. ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac
(2*n+1)/(n+2)
1/n
0 Your answer is not algebraically equivalent to the correct answer. You must have done something wrong. ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac
(2*n)/(n*(n+2))
(2*n)/(n*(n+3))
0 Your answer is not algebraically equivalent to the correct answer. You must have done something wrong. ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac
(x-1)/(x^2-1)
1/(x+1)
1 ATSingleFrac_true.
Fractions within fractions
SingleFrac
(1/2)/(3/4)
2/3
SingleFrac
(x-2)/4/(2/x^2)
(x-2)*x^2/8
SingleFrac
1/(1-1/x)
x/(x-1)
SingleFrac
(1+1/a)/a
(1+a)/a^2
SingleFrac
a/(1+1/a)
a^2/(1+a)
SingleFrac
(1+2*b/a)/c
(a+2*b)/(a*c)
SingleFrac
c/(1+2*b/a)
a*c/(a+2*b)
SingleFrac
a*c/(a+2*b)
a*c/(a+2*b)
1 ATSingleFrac_true.
Negative cases
SingleFrac
-1/2
-1/2
1 ATSingleFrac_true.
SingleFrac
-1/2
-1/3
0 Your answer is not algebraically equivalent to the correct answer. You must have done something wrong. ATSingleFrac_true. ATSingleFrac_ret_exp.
SingleFrac
-(1/2)
-1/2
1 ATSingleFrac_true.
SingleFrac
-a/b
-a/b
1 ATSingleFrac_true.
SingleFrac
(-a)/b
-a/b
1 ATSingleFrac_true.
SingleFrac
a/(-b)
-a/b
1 ATSingleFrac_true.
SingleFrac
-(a/b)
-a/b
1 ATSingleFrac_true.
SingleFrac
-(1/(n-1))
1/(1-n)
1 ATSingleFrac_true.
SingleFrac
a/(-1-1/a)
-a^2/(1+a)
SingleFrac
((sqrt(5))^3 +6)/15
((sqrt(5))^3 +6)/15
1 ATSingleFrac_true.
SingleFrac
1/(1-sqrt(2))
1/(1-sqrt(2))
1 ATSingleFrac_true.
SingleFrac
((sqrt(5))^3+6)/15
((sqrt(5))^3+6)/15
1 ATSingleFrac_true.
SingleFrac
(5^(3/2)+6)/15
((sqrt(5))^3+6)/15
1 ATSingleFrac_true.

## PartFrac

Test
?
Student response
Opt
Mark
CAS errors
Feedback
PartFrac
1/0
3*x^2
PartFrac
1/0
3*x^2
x
PartFrac
0
0
1/0
PartFrac
0
1/0
x
PartFrac
1/n=0
1/n
n
0 Your answer should be an expression, not an equation, inequality, list, set or matrix. ATPartFrac_SA_not_expression.
PartFrac
1/n
{1/n}
n
Basic tests
PartFrac
1/m
1/n
n
0 The variables in your answer are different to those of the question, please check them. ATPartFrac_diff_variables.
PartFrac
2/(x+1)-1/(x+2)
s/((s+1)*(s+2))
s
0 The variables in your answer are different to those of the question, please check them. ATPartFrac_diff_variables.
PartFrac
1/n
1/n
n
1 ATPartFrac_true.
PartFrac
n^3/(n-1)
n^3/(n-1)
n
0 ATPartFrac_false_factor.
PartFrac
1+n+n^2+1/(n-1)
n^3/(n-1)
n
1 ATPartFrac_true.
PartFrac
1+n+n^2-1/(1-n)
n^3/(n-1)
n
1 ATPartFrac_true.
Distinct linear factors in denominator
PartFrac
1/(n+1)-1/n
1/(n+1)-1/n
n
1 ATPartFrac_true.
PartFrac
1/(n+1)+1/(1-n)
1/(n+1)-1/(n-1)
n
1 ATPartFrac_true.
PartFrac
1/(2*(n-1))-1/(2*(n+1))
1/((n-1)*(n+1))
n
1 ATPartFrac_true.
PartFrac
1/(2*(n+1))-1/(2*(n-1))
1/((n-1)*(n+1))
n
0 Your answer as a single fraction is $$-\frac{1}{\left(n-1\right)\cdot \left(n+1\right)}$$ ATPartFrac_ret_expression.
PartFrac
-9/(x-2) + -9/(x+1)
-9/(x-2) + -9/(x+1)
x
1 ATPartFrac_true.
PartFrac
1/(x+1) + 1/(x+2)
2/(x+1) + 1/(x+2)
x
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
1/(x+1) + 1/(x+2)
1/(x+1) + 2/(x+2)
x
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
1/(x+1) + 1/(x+2)
1/(x+3) + 1/(x+2)
x
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.
Repeated linear factors in denominator
PartFrac
(9*y-8)/(y-4)^2
(9*y-8)/(y-4)^2
y
0 ATPartFrac_false_factor.
PartFrac
9/(y-4)+28/(y-4)^2
(9*y-8)/(y-4)^2
y
1 ATPartFrac_true.
PartFrac
(-5/(x+3))+(16/(x+3)^2)-(2/(x+
2))+4
(-5/(x+3))+(16/(x+3)^2)-(2/(x+
2))+4
x
1 ATPartFrac_true.
PartFrac
(3*x^2-5)/((x-4)^2*x)
(3*x^2-5)/((x-4)^2*x)
x
0 ATPartFrac_false_factor.
PartFrac
-4/(16*x)+53/(16*(x-4))+43/(4*
(x-4)^2)
(3*x^2-5)/((x-4)^2*x)
x
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
-5/(16*x)+53/(16*(x-4))+43/(4*
(x-4)^2)
(3*x^2-5)/((x-4)^2*x)
x
1 ATPartFrac_true.
PartFrac
(5*x+6)/((x+1)*(x+5)^2)
(5*x+6)/((x+1)*(x+5)^2)
x
0 ATPartFrac_false_factor.
PartFrac
-1/(16*(x+5))+19/(4*(x+5)^2)+1
/(16*(x+1))
(5*x+6)/((x+1)*(x+5)^2)
x
1 ATPartFrac_true.
PartFrac
5/(x*(x+3)*(5*x-2))
5/(x*(x+3)*(5*x-2))
x
0 ATPartFrac_false_factor.
PartFrac
125/(34*(5*x-2))+5/(51*(x+3))-
5/(6*x)
5/(x*(x+3)*(5*x-2))
x
1 ATPartFrac_true.
PartFrac
-4/(16*x)+1/(2*(x-1))-1/(8*(x-
1)^2)
(3*x^2-5)/((4*x-4)^2*x)
x
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
-5/(16*x)+1/(2*(x-1))-1/(8*(x-
1)^2)
(3*x^2-5)/((4*x-4)^2*x)
x
1 ATPartFrac_true.
PartFrac
1/(x-1)-(x+1)/(x^2+1)
2/((x-1)*(x^2+1))
x
1 ATPartFrac_true.
PartFrac
1/(2*x-2)-(x+1)/(2*(x^2+1))
1/((x-1)*(x^2+1))
x
1 ATPartFrac_true.
PartFrac
1/(2*(x-1))+x/(2*(x^2+1))
1/((x-1)*(x^2+1))
x
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.
PartFrac
(2*x+1)/(x^2+1)-2/(x-1)
(2*x+1)/(x^2+1)-2/(x-1)
x
1 ATPartFrac_true.
2 answers to the same question
PartFrac
3/(x+1) + 3/(x+2)
3*(2*x+3)/((x+1)*(x+2))
x
1 ATPartFrac_true.
PartFrac
3*(1/(x+1) + 1/(x+2))
3*(2*x+3)/((x+1)*(x+2))
x
1 ATPartFrac_true.
Algebraically equivalent, but numerators of same order than denominator, i.e. not in partial fraction form.
PartFrac
3*x*(1/(x+1) + 2/(x+2))
-12/(x+2)-3/(x+1)+9
x
0 ATPartFrac_false_factor.
PartFrac
(3*x+3)*(1/(x+1) + 2/(x+2))
9-6/(x+2)
x
0 ATPartFrac_false_factor.
PartFrac
n/(2*n-1)-(n+1)/(2*n+1)
1/(4*n-2)-1/(4*n+2)
n
0 ATPartFrac_false_factor.
PartFrac
10/(x+3) - 2/(x+2) + x -2
(x^3 + 3*x^2 + 4*x +2)/((x+2)*
(x+3))
x
1 ATPartFrac_true.
PartFrac
2*x+1/(x+1)+1/(x-1)
2*x^3/(x^2-1)
x
1 ATPartFrac_true.
Simple mistakes
PartFrac
1/(n*(n-1))
1/(n*(n-1))
n
0 ATPartFrac_false_factor.
PartFrac
((1-x)^4*x^4)/(x^2+1)
((1-x)^4*x^4)/(x^2+1)
x
0 ATPartFrac_false_factor.
PartFrac
1/(n-1)-1/n^2
1/((n+1)*n)
n
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
1/(n-1)-1/n
1/(n-1)+1/n
n
0 Your answer as a single fraction is $$\frac{1}{\left(n-1\right)\cdot n}$$ ATPartFrac_ret_expression.
PartFrac
1/(x+1)-1/x
1/(x-1)+1/x
x
0 Your answer as a single fraction is $$-\frac{1}{x\cdot \left(x+1\right)}$$ ATPartFrac_ret_expression.
PartFrac
1/(n*(n+1))+1/n
2/n-1/(n+1)
n
0 ATPartFrac_false_factor.
Too many parts in the partial fraction
PartFrac
s/((s+1)^2) + s/(s+2) - 1/(s+1
)
s/((s+1)*(s+2))
s
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
s/(s+2) - 1/(s+1)
s/((s+1)*(s+2)*(s+3))
s
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.
PartFrac
(3*x^2-5)/((4*x-4)^2*x)
(3*x^2-5)/((4*x-4)^2*x)
x
0 ATPartFrac_false_factor.

## Diff

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Diff
1/0
3*x^2
Diff
0
1/0
(x
-1 TEST_FAILED 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
1/0
3*x^2
x
-1 ATDiff_STACKERROR_SAns.
Diff
0
1/0
x
-1 ATDiff_STACKERROR_TAns.
Diff
0
0
1/0
-1 ATDiff_STACKERROR_Opt.
Basic tests
Diff
3*x^2
3*x^2
x
1 ATDiff_true.
Diff
3*X^2
3*x^2
x
0 ATDiff_var_SB_notSA.
Diff
x^4/4
3*x^2
x
0 It looks like you have integrated instead! ATDiff_int.
Diff
x^4/4+1
3*x^2
x
0 It looks like you have integrated instead! ATDiff_int.
Diff
x^4/4+c
3*x^2
x
0 It looks like you have integrated instead! ATDiff_int.
Diff
y=x^4/4
x^4/4
x
0 Your answer should be an expression, not an equation, inequality, list, set or matrix. ATDiff_SA_not_expression.
Diff
x^4/4
y=x^4/4
x
0
Diff
y=x^4/4
y=x^4/4
x
0 Your answer should be an expression, not an equation, inequality, list, set or matrix. ATDiff_SA_not_expression.
Diff
6000*(x-a)^5999
6000*(x-a)^5999
x
1 ATDiff_true.
Diff
5999*(x-a)^5999
6000*(x-a)^5999
x
0
Variable mismatch tests
Diff
y^2-2*y+1
x^2-2*x+1
x
0 ATDiff_var_SB_notSA.
Diff
x^2-2*x+1
y^2-2*y+1
x
0 ATDiff_var_SA_notSB.
Diff
y^2+2*y+1
x^2-2*x+1
z
0 ATDiff_var_notSASB_SAnceSB.
Diff
x^4/4
3*x^2
y
0
Edge cases
Diff
e^x+c
e^x
x
0 It looks like you have integrated instead! ATDiff_int.
Diff
e^x+2
e^x
x
0 It looks like you have integrated instead! ATDiff_int.
Diff
n*x^n
n*x^(n-1)
x
Diff
n*x^n
(assume(n>0), n*x^(n-1))
x
0

## Int

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Int
1/0
1
Int
1/0
1
x
-1 ATInt_STACKERROR_SAns.
Int
1
1/0
x
-1 ATInt_STACKERROR_TAns.
Int
0
0
1/0
-1 ATInt_STACKERROR_Opt.
Int
0
0
[x,1/0]
-1 ATInt_STACKERROR_Opt.
Int
0
0
[x,NOCONST,1/0]
-1 ATInt_STACKERROR_Opt.
Basic tests
Int
x^3/3
x^3/3
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
x^3/3+1
x^3/3
x
0 You need to add a constant of integration. This should be an arbitrary constant, not a number. ATInt_const_int.
Int
x^3/3+c
x^3/3
x
1 ATInt_true.
Int
x^3/3+c+1
x^3/3
x
1 ATInt_true.
Int
x^3/3+3*c
x^3/3
x
1 ATInt_true.
Int
(x^3+c)/3
x^3/3
x
1 ATInt_true.
Int
x^3/3-c
x^3/3
x
1 ATInt_true.
Int
x^3/3+c+k
x^3/3
x
Int
x^3/3+c^2
x^3/3
x
Int
x^3/3*c
x^3/3
x
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
X^3/3+c
x^3/3
x
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
sin(2*x)
x^3/3
x
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
x^2/2-2*x+2+c
(x-2)^2/2
x
1 ATInt_true.
Int
(t-1)^5/5+c
(t-1)^5/5
t
1 ATInt_true.
Int
(v-1)^5/5+c
(v-1)^5/5
v
1 ATInt_true.
Int
cos(2*x)/2+1+c
cos(2*x)/2
x
1 ATInt_true.
Int
(x-a)^6001/6001+c
(x-a)^6001/6001
x
1 ATInt_true.
Int
(x-a)^6001/6001
(x-a)^6001/6001
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
6000*(x-a)^5999
(x-a)^6001/6001
x
0 It looks like you have differentiated instead! ATInt_diff.
Int
4*%e^(4*x)/(%e^(4*x)+1)
log(%e^(4*x)+1)+c
x
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.
Int
x^3/3+c
x^3/3+c
x
1 ATInt_true.
Int
x^2/2-2*x+2+c
(x-2)^2/2+k
x
1 ATInt_true.
The teacher condones lack of constant, or numerical constant
Int
x^3/3
x^3/3
[x,NOCONST]
1 ATInt_const_condone.
Int
x^3/3+c
x^3/3
[x,NOCONST]
1 ATInt_true.
Int
x^2/2-2*x+2
(x-2)^2/2+k
[x,NOCONST]
1 ATInt_const_condone.
Int
x^3/3+1
x^3/3
[x,NOCONST]
1 ATInt_const_int_condone.
Int
x^3/3+c^2
x^3/3
[x,NOCONST]
Int
n*x^n
n*x^(n-1)
x
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
n*x^n
(assume(n>0), n*x^(n-1))
x
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
exp(x)+c
exp(x)
x
1 ATInt_true.
Int
exp(x)
exp(x)
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
exp(x)
exp(x)
[x,NOCONST]
1 ATInt_const_condone.
Student differentiates by mistake
Int
2*x
x^3/3
x
0 It looks like you have differentiated instead! ATInt_diff.
Int
2*x+c
x^3/3
x
0 It looks like you have differentiated instead! ATInt_diff.
Sloppy logs (teacher ignores abs(x) )
Int
ln(x)
ln(x)
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
ln(x)
ln(x)
[x,NOCONST]
1 ATInt_const_condone.
Int
ln(x)+c
ln(x)+c
x
1 ATInt_true_equiv.
Int
ln(k*x)
ln(x)+c
x
1 ATInt_true_equiv.
Fussy logs (teacher uses abs(x) )
Int
ln(x)
ln(abs(x))+c
x
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
ln(x)+c
ln(abs(x))+c
x
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
ln(x)
ln(abs(x))+c
[x, NOCONST]
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
ln(abs(x))
ln(abs(x))+c
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
ln(abs(x))+c
ln(abs(x))+c
x
1 ATInt_true_equiv.
Int
ln(k*x)
ln(abs(x))+c
x
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
ln(k*abs(x))
ln(abs(x))+c
x
1 ATInt_true_equiv.
Int
ln(abs(k*x))
ln(abs(x))+c
x
1 ATInt_true_equiv.
Teacher uses ln(k*abs(x))
Int
ln(x)
ln(k*abs(x))
x
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
ln(x)+c
ln(k*abs(x))
x
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
ln(abs(x))
ln(k*abs(x))
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
ln(abs(x))+c
ln(k*abs(x))
x
1 ATInt_true_equiv.
Int
ln(k*x)
ln(k*abs(x))
x
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
ln(k*abs(x))
ln(k*abs(x))
x
1 ATInt_true_equiv.
Other logs
Int
ln(x)+ln(a)
ln(k*abs(x+a))
x
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
log(x)^2-2*log(c)*log(x)+k
ln(c/x)^2
x
Int
log(x)^2-2*log(c)*log(x)+k
ln(abs(c/x))^2
x
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
c-(log(2)-log(x))^2/2
-1/2*log(2/x)^2
x
1 ATInt_true_equiv.
Int !
ln(abs(x+3))/2+c
ln(abs(2*x+6))/2+c
x
Two logs
Int
log(abs(x-3))+log(abs(x+3))
log(abs(x-3))+log(abs(x+3))
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
log(abs(x-3))+log(abs(x+3))+c
log(abs(x-3))+log(abs(x+3))
x
1 ATInt_true_equiv.
Int
log(abs(x-3))+log(abs(x+3))
log(x-3)+log(x+3)
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
log(abs(x-3))+log(abs(x+3))+c
log(x-3)+log(x+3)
x
1 ATInt_true_equiv.
Int
log(x-3)+log(x+3)
log(x-3)+log(x+3)
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
log(x-3)+log(x+3)+c
log(x-3)+log(x+3)
x
1 ATInt_true_equiv.
Int
log(x-3)+log(x+3)
log(abs(x-3))+log(abs(x+3))
x
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
log(x-3)+log(x+3)+c
log(abs(x-3))+log(abs(x+3))
x
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
log(abs((x-3)*(x+3)))+c
log(abs(x-3))+log(abs(x+3))
x
1 ATInt_true_equiv.
Int
log(abs((x^2-9)))+c
log(abs(x-3))+log(abs(x+3))
x
Int
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 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
-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 ATInt_true_equiv.
Int
-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 ATInt_true_equiv.
Inconsistent log(abs())
Int
log(abs(x-3))+log((x+3))+c
log(x-3)+log(x+3)
x
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
log((v-3))+log(abs(v+3))+c
log(v-3)+log(v+3)
v
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
log((x-3))+log(abs(x+3))
log(x-3)+log(x+3)
x
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
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 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
2*(sqrt(t)-5)-10*log((sqrt(t)-
5))+c
2*(sqrt(t)-5)-10*log((sqrt(t)-
5))+c
t
1 ATInt_true_equiv.
Int
2*(sqrt(t))-10*log((sqrt(t)-5)
)+c
2*(sqrt(t)-5)-10*log((sqrt(t)-
5))+c
t
1 ATInt_true_differentconst.
Int
2*(sqrt(t)-5)-10*log((sqrt(t)-
5))+c
2*(sqrt(t)-5)-10*log(abs(sqrt(
t)-5))+c
t
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
2*(sqrt(t))-10*log(abs(sqrt(t)
-5))+c
2*(sqrt(t)-5)-10*log(abs(sqrt(
t)-5))+c
t
1 ATInt_true_differentconst.
Trig
Int
2*sin(x)*cos(x)
sin(2*x)+c
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
2*sin(x)*cos(x)+k
sin(2*x)+c
x
1 ATInt_true.
Int
-2*cos(3*x)/3-3*cos(2*x)/2
-2*cos(3*x)/3-3*cos(2*x)/2+c
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
-2*cos(3*x)/3-3*cos(2*x)/2+1
-2*cos(3*x)/3-3*cos(2*x)/2+c
x
0 You need to add a constant of integration. This should be an arbitrary constant, not a number. ATInt_const_int.
Int
-2*cos(3*x)/3-3*cos(2*x)/2+c
-2*cos(3*x)/3-3*cos(2*x)/2+c
x
1 ATInt_true.
Int
(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 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
(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 You need to add a constant of integration. This should be an arbitrary constant, not a number. ATInt_const_int.
Int
(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 ATInt_true.
Int
tan(x)-x+c
tan(x)-x
x
1 ATInt_true.
Note the difference in feedback here, generated by the options.
Int
((5*%e^7*x-%e^7)*%e^(5*x))
((5*%e^7*x-%e^7)*%e^(5*x))/25+
c
x
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
((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 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
log(x-3)/6-log(x+3)/6+c
log(x-3)/6-log(x+3)/6
x
1 ATInt_true_equiv.
Int
asinh(x)
ln(x+sqrt(x^2+1))
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
asinh(x)+c
ln(x+sqrt(x^2+1))
x
1 ATInt_true.
Int
-acoth(x/3)/3
log(x-3)/6-log(x+3)/6
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
-acoth(x/3)/3
log(x-3)/6-log(x+3)/6
[x, NOCONST]
1 ATInt_true.
Int
-acoth(x/3)/3+c
log(x-3)/6-log(x+3)/6
x
1 ATInt_true.
Int
-acoth(x/3)/3+c
log(abs(x-3))/6-log(abs(x+3))/
6
x
1 ATInt_true.
Int
log(x-a)/(2*a)-log(x+a)/(2*a)+
c
log(x-a)/(2*a)-log(x+a)/(2*a)
x
1 ATInt_true_equiv.
Int
-acoth(x/a)/a+c
log(x-a)/(2*a)-log(x+a)/(2*a)
x
1 ATInt_true.
Int
-acoth(x/a)/a+c
log(abs(x-a))/(2*a)-log(abs(x+
a))/(2*a)
x
1 ATInt_true.
Int
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 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
log(x-3)/6-log(x+3)/6+c
-acoth(x/3)/3
x
1 ATInt_true.
Int
log(abs(x-3))/6-log(abs(x+3))/
6+c
-acoth(x/3)/3
x
1 ATInt_true.
Int
log(x-3)/6-log(x+3)/6
-acoth(x/3)/3
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
atan(2*x-3)+c
atan(2*x-3)
x
1 ATInt_true.
Int
atan((x-2)/(x-1))+c
atan(2*x-3)
x
1 ATInt_true.
Int
atan((x-2)/(x-1))
atan(2*x-3)
x
0 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.
Int
atan((x-1)/(x-2))
atan(2*x-3)
x
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 !
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
-3 You need to add a constant of integration, otherwise this appears to be correct. Well done. ATInt_const.

## GT

Test
?
Student response
Opt
Mark
CAS errors
Feedback
GT
1/0
1
GT
1
1/0
GT
1
1
0 ATGT_false.
GT
2
1
1 ATGT_true.
GT
1
2.1
0 ATGT_false.
GT
pi
3
1 ATGT_true.
GT
pi+2
5
1 ATGT_true.
Infinity
GT
-inf
0
0 Not number
GT
inf
0
0 Not number

## GTE

Test
?
Student response
Opt
Mark
CAS errors
Feedback
GTE
1/0
1
GTE
1
1/0
GTE
1
1
1 ATGTE_true.
GTE
2
1
1 ATGTE_true.
GTE
1
2.1
0 ATGTE_false.
GTE
pi
3
1 ATGTE_true.
GTE
pi+2
5
1 ATGTE_true.

## NumRelative

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Basic tests
NumRelative
1/0
0
NumRelative
0
1/0
NumRelative
0
0
1/0
NumRelative
0
(x
NumRelative
1.5
1.5
x
NumRelative
1
0
(x
0
NumRelative
x=1.5
1.5
0 Your answer should be a floating point number, but is not. ATNumerical_SA_not_number.
NumRelative
1.5
x=1.5
No option, so 5%
NumRelative
1.1
1
0
NumRelative
1.05
1
1
NumRelative
0.95
1
1
NumRelative
0.949
1
0
NumRelative
1.05e33
1e33
1
NumRelative
1.06e33
1e33
0
NumRelative
0.95e33
1e33
1
NumRelative
0.949e33
1e33
0
NumRelative
1.05e-33
1e-33
1
NumRelative
1.06e-33
1e-33
0
NumRelative
0.95e-33
1e-33
1
NumRelative
0.949e-33
1e-33
0
Remove display dp etc.
NumRelative
1
displaydp(1.05,2)
0.1
1
NumRelative
1000
displaysci(1.05,2,3)
0.1
1
Options passed
NumRelative
1.05
1
0.1
1
NumRelative
1.05
3
0.1
0
NumRelative
3.14
pi
0.001
1
Infinity
NumRelative
inf
0
0 Your answer should be a floating point number, but is not. ATNumerical_SA_not_number.
Lists
NumRelative
1
[1,2]
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
[1,2]
[1,2,3]
0 Your list should have $$3$$ elements, but it actually has $$2$$. ATNumerical_wronglen.
NumRelative
[1,2]
[1,2]
1
NumRelative
[3.141,1.414]
[pi,sqrt(2)]
1
NumRelative
[3,1.414]
[pi,sqrt(2)]
0.01
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
[3,1.414]
{pi,sqrt(2)}
0.01
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
{1.414,3.1}
{significantfigures(pi,6),sqrt
(2)}
0.01
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
{1.414,3.1}
{pi,sqrt(2)}
0.1
1

## NumAbsolute

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Basic tests
NumAbsolute
1/0
0
NumAbsolute
0
1/0
NumAbsolute
0
0
1/0
NumAbsolute
0
(x
NumAbsolute
1
0
(x
0
No option, so 5%
NumAbsolute
1.1
1
0
NumAbsolute
1.05
1
1
Options passed
NumAbsolute
1.05
1
0.1
1
NumAbsolute
1.05
3
0.1
0
NumAbsolute
3.14
pi
0.001
0
NumAbsolute
1.41e-2
1.41e-2
0.0001
1
NumAbsolute
0.0141
1.41e-2
0.0001
1
NumAbsolute
0.00141
0.00141
0.0001
1
NumAbsolute
0.00141
1.41*10^-3
0.0001
1
NumAbsolute
1.41*10^-3
1.41*10^-3
0.0001
1
NumAbsolute
[3.141,1.414]
[pi,sqrt(2)]
0.01
1
NumAbsolute
[3,1.414]
[pi,sqrt(2)]
0.01
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
[3,1.414]
{pi,sqrt(2)}
0.01
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
{1.414,3.1}
{significantfigures(pi,6),sqrt
(2)}
0.01
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
{1,1.414,3.1,2}
{1,2,pi,sqrt(2)}
0.1
1

## NumSigFigs

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Basic tests
NumSigFigs
3.141
3.1415927
NumSigFigs
1/0
3
3
-1 ATNumSigFigs_STACKERROR_SAns.
NumSigFigs
0
1/0
3
-1 ATNumSigFigs_STACKERROR_TAns.
NumSigFigs
0
0
1/0
-1 ATNumSigFigs_STACKERROR_Opt.
NumSigFigs
0
1
(
-1 TEST_FAILED 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
(
1
1
NumSigFigs
1
3
pi
NumSigFigs
1
3
[3,x]
NumSigFigs
1
3
[1,2,3]
NumSigFigs
1
3
NumSigFigs
pi
pi
4
0 Your answer should be a decimal number, but is not! ATNumSigFigs_NotDecimal.
Edge cases
NumSigFigs
0
0
2
NumSigFigs
0
0
1
1
NumSigFigs
0.0
0
1
1
NumSigFigs
0.0
0
2
NumSigFigs
0
0.0
2
NumSigFigs
0.0
0.0
2
NumSigFigs
0.00
0.00
2
1
Large numbers
NumSigFigs
5.4e21
5.3e21
2
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
5.3e21
5.3e21
2
1
NumSigFigs
5.3e22
5.3e22
2
1
NumSigFigs
5.3e20
5.3e22
2
0 ATNumSigFigs_VeryInaccurate.
NumSigFigs
6.02214086e23
6.02214086e23
9
1
NumSigFigs
6.0221409e23
6.02214086e23
9
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
6.02214087e23
6.02214086e23
9
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
6.02214085e23
6.02214086e23
9
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
5.3910632e-44
5.3910632e-44
8
1
NumSigFigs
5.391063e-44
5.3910632e-44
8
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
5.3910631e-44
5.3910632e-44
8
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
5.3910633e-44
5.3910632e-44
8
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
1.61622938e-35
1.61622938e-35
9
1
NumSigFigs
1.6162294e-35
1.61622938e-35
9
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
1.61622939e-35
1.61622938e-35
9
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
1.61622937e-35
1.61622938e-35
9
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
1.2345e82
1.2345e82
5
1
NumSigFigs
1.2346e82
1.2345e82
5
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
1.2344e82
1.2345e82
5
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
1.234
4
1
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
3.141
3.1415927
3
NumSigFigs
3.141
3.1415927
4
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
3.146
3.1415927
4
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
3.147
3.1415927
4
0 ATNumSigFigs_VeryInaccurate.
NumSigFigs
3.142
3.1415927
4
1
NumSigFigs
3.142
pi
4
1
NumSigFigs
3141
3.1415927
4
0 ATNumSigFigs_VeryInaccurate.
NumSigFigs
0.00123
0.001234567
3
1
NumSigFigs
1.23e-3
0.001234567
3
1
NumSigFigs
138*10^-3
138*10^-3
3
1
NumSigFigs
-138*10^-3
-138*10^-3
3
1
NumSigFigs
138*10^-3
-138*10^-3
3
NumSigFigs
1.38*10^-1
138*10^-3
3
1
NumSigFigs
1.24e-3
0.001234567
3
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
1.235e-3
0.001234567
4
1
NumSigFigs
1000
999
2
1 ATNumSigFigs_WithinRange.
NumSigFigs
1E3
999
2
NumSigFigs
-100
-149
1
1
NumSigFigs
-0.05
-0.0499
1
1
NumSigFigs
-(0.05)
-0.0499
1
1
NumSigFigs
1170
1174.34
3
1
NumSigFigs
61300
61250
3
1
Previous tricky case
NumSigFigs
0.1667
0.1667
4
1
NumSigFigs
0.1666
0.1667
4
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
0.1663
0.1667
4
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
0.1662
0.1667
4
0 ATNumSigFigs_VeryInaccurate.
NumSigFigs
0.166
0.1667
4
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATNumSigFigs_VeryInaccurate.
NumSigFigs
0.16667
0.1667
4
Negative numbers
NumSigFigs
-3.141
-3.1415927
4
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
-3.141
-3.1415927
3
NumSigFigs
-3.141
-3.1415927
4
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
-3.142
-3.1415927
4
1
NumSigFigs
3.142
-3.1415927
4
NumSigFigs
-3.142
3.1415927
4
NumSigFigs
-3.149
3.1415927
4
NumSigFigs
2.15
75701719/35227192
3
1
NumSigFigs
0.0499
0.04985
3
1
NumSigFigs
0.0498
0.04985
3
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
0.0498
0.04975
3
1
NumSigFigs
0.0497
0.04975
3
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
0.0499
0.0498
3
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
1.5
1.500
3
NumSigFigs
1.50
1.500
3
1
NumSigFigs
1.500
1.500
3
NumSigFigs
245.0
245
3
Too few digits
NumSigFigs
180
178.35
3
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
33
33.1558
3
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
3.142
3.1415927
[4,3]
1
NumSigFigs
3.143
3.1415927
[4,3]
1
NumSigFigs
3.150
3.1415927
[4,3]
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
3.211
3.1415927
[4,3]
0 ATNumSigFigs_VeryInaccurate.
NumSigFigs
3.1416
3.1415927
[4,3]
NumSigFigs
0.1666
0.1667
[4,3]
1
NumSigFigs
180
178.35
[3,1]
1 ATNumSigFigs_WithinRange.
NumSigFigs
33
33.1558
[3,1]
NumSigFigs
1.500
1.5
[3,1]
NumSigFigs
245.0
245
[3,1]
NumSigFigs
12345.7
12345.654321
[6,6]
1
NumSigFigs
12345.7
12345.654321
[6,3]
1
NumSigFigs
12300.0
12345.654321
[6,3]
1
NumSigFigs
12400.0
12345.654321
[6,3]
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
13500.0
12345.654321
[6,3]
0 ATNumSigFigs_VeryInaccurate.
NumSigFigs
12000.0
12345.654321
[6,2]
1
NumSigFigs
13000.0
12345.654321
[6,2]
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
11000.0
12345.654321
[6,2]
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
0.0010
0
[1,0]
NumSigFigs
0.0010
0
[2,0]
1
NumSigFigs
0.0010
0
[3,0]
NumSigFigs
0.001
0
[1,0]
1
NumSigFigs
0.001
0
[2,0]
NumSigFigs
0.00100
null
[2,0]
NumSigFigs
0.00100
null
[3,0]
1
NumSigFigs
0.00100
null
[4,0]
NumSigFigs
5.00
null
[2,0]
NumSigFigs
5.00
null
[3,0]
1
NumSigFigs
5.00
null
[4,0]
NumSigFigs
100
0
[1,0]
1
NumSigFigs
100
0
[2,0]
1 ATNumSigFigs_WithinRange.
NumSigFigs
100
0
[3,0]
1 ATNumSigFigs_WithinRange.
NumSigFigs
100
0
[4,0]
NumSigFigs
10.0
0
[2,0]
NumSigFigs
10.0
0
[3,0]
1
NumSigFigs
10.0
0
[4,0]
NumSigFigs
0
0
[1,0]
1
NumSigFigs
0
0
[2,0]
NumSigFigs
0.00
0
[1,0]
NumSigFigs
0.00
0
[2,0]
1
NumSigFigs
0.00
0
[3,0]
NumSigFigs
0.00
0
[4,0]
Condone too many sfs.
NumSigFigs
8.250
8.250
[4,-1]
1
NumSigFigs
8.25
8.250
[4,-1]
NumSigFigs
8.250000
8.250
[4,-1]
1
NumSigFigs
8.250434
8.250
[4,-1]
1
NumSigFigs
82.4
82
[2,-1]
1
NumSigFigs
82.5
82
[2,-1]
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
83
82
[2,-1]
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
0.1430
1/7
[4,-1]
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
0.1429
1/7
[4,-1]
1
NumSigFigs
0.1428
1/7
[4,-1]
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
0.143
1/7
[4,-1]
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
0.14284
1/7
[4,-1]
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
0.14285
1/7
[4,-1]
1
NumSigFigs
0.14286
1/7
[4,-1]
1
NumSigFigs
0.14291
1/7
[4,-1]
1
NumSigFigs
0.14294
1/7
[4,-1]
1
NumSigFigs
0.14295
1/7
[4,-1]
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
0.142
1/7
[2,-1]
1
NumSigFigs
0.14290907676
1/7
[2,-1]
1
NumSigFigs
0.143
1/7
[2,-1]
1
NumSigFigs
0.1433333
1/7
[2,-1]
1
NumSigFigs
0.144
1/7
[2,-1]
1
NumSigFigs
0.145
1/7
[2,-1]
1
NumSigFigs
0.146
1/7
[2,-1]
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
1.279
ev(lg(19),lg=logbasesimp)
4
1
NumSigFigs
3.14
pi
3
1
NumSigFigs
3.15
pi
3
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
1.73205
sqrt(3)
6
1
No support for matrices!
NumSigFigs
matrix([0.33,1],[1,1])
matrix([0.333,1],[1,1])
2
-1 Your answer should be a decimal number, but is not! ATNumSigFigs_NotDecimal.
NumSigFigs
3.1415
matrix([0.333,1],[1,1])
2
-1 TEST_FAILED 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
1.50
dispsf(1.500,3)
3
1
NumSigFigs
1.50
dispdp(1.500,3)
3
1

## NumDecPlaces

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Basic tests
NumDecPlaces
1/0
3
2
-1 ATNumDecPlaces_STACKERROR_SAns.
NumDecPlaces
0.1
1/0
2
-1 ATNumDecPlaces_STACKERROR_TAns.
NumDecPlaces
0.1
0
1/0
-1 ATNumDecPlaces_STACKERROR_Opt.
NumDecPlaces
0.1
1
x
-1 For ATNumDecPlaces the test option must be a positive integer, in fact "$$x$$" was received. ATNumDecPlaces_OptNotInt.
NumDecPlaces
0.1
1
-1
-1 For ATNumDecPlaces the test option must be a positive integer, in fact "$$-1$$" was received. ATNumDecPlaces_OptNotInt.
NumDecPlaces
0.1
1
0
-1 For ATNumDecPlaces the test option must be a positive integer, in fact "$$0$$" was received. ATNumDecPlaces_OptNotInt.
NumDecPlaces
0.1
1
(
-1 TEST_FAILED 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
(
1
1
Student's answer not a floating point number
NumDecPlaces
x
3.143
2
0 Your answer must be a floating point number, but is not. ATNumDecPlaces_SA_Not_num.
NumDecPlaces
pi
3.000
3
0 Your answer must be a floating point number, but is not. ATNumDecPlaces_SA_Not_num.
Right number of places
NumDecPlaces
3.14
3.143
2
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces
3.14
3.14
2
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces
3.140
3.140
3
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces
3141.5972
3141.5972
4
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces
4.14
3.14
2
0 ATNumDecPlaces_Correct. ATNumDecPlaces_Not_equiv.
NumDecPlaces
3.1416
pi
4
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
NumDecPlaces
-7.3
-7.3
1
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
Wrong number of places
NumDecPlaces
3.14
3.143
1
0 Your answer has been given to the wrong number of decimal places. ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Equiv.
NumDecPlaces
3.14
3.143
1
0 Your answer has been given to the wrong number of decimal places. ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Equiv.
NumDecPlaces
3.14
3.140
3
0 Your answer has been given to the wrong number of decimal places. ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Equiv.
NumDecPlaces
7.000
7
4
0 Your answer has been given to the wrong number of decimal places. ATNumDecPlaces_Wrong_DPs. ATNumDecPlaces_Equiv.
NumDecPlaces
7.0000
7
4
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
Both wrong DPs and inaccurate.
NumDecPlaces
8.0000
7
3
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
4.000
3.99999
3
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.
Teacher uses displaydp
NumDecPlaces
0.10
displaydp(0.1,2)
2
1 ATNumDecPlaces_Correct. ATNumDecPlaces_Equiv.

## NumDecPlacesWrong

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Basic tests
NumDecPlacesWrong
1/0
3
2
-1 ATNumDecPlacesWrong_STACKERROR_SAns.
NumDecPlacesWrong
0.1
1/0
2
-1 ATNumDecPlacesWrong_STACKERROR_TAns.
NumDecPlacesWrong
0.1
0
1/0
-1 ATNumDecPlacesWrong_STACKERROR_Opt.
NumDecPlacesWrong
0.1
0
x
-1 For ATNumDecPlacesWrong the test option must be a positive integer, in fact "$$x$$" was received. ATNumDecPlacesWrong_OptNotInt.
NumDecPlacesWrong
x^2
1234
4
0 Your answer must be a floating point number, but is not. ATNumDecPlacesWrong_SA_Not_num.
NumDecPlacesWrong
1234.5
x^2
4
0 ATNumDecPlacesWrong_Tans_Not_Num.
NumDecPlacesWrong
3.141
31.41
4
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
3.141
31.14
4
0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong
pi
31.14
4
0 Your answer must be a floating point number, but is not. ATNumDecPlacesWrong_SA_Not_num.
NumDecPlacesWrong
0.1234
1234
4
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
0.1235
1234
4
0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong
0.0001234
1234
4
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
0.0001235
1234
4
0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong
0.1233
1234
3
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
0.1243
1234
3
0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong
0.1230
1239
3
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
0.1240
1239
3
0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong
1230
1239
3
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
2230
1239
3
0 ATNumDecPlacesWrong_Wrong.
NumDecPlacesWrong
0.100
1.00
3
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
0.1000
1.00
3
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
0.1001
1.001
3
1 ATNumDecPlacesWrong_Correct.
Condone lack of trailing zeros
NumDecPlacesWrong
0.100
1.0
4
1 ATNumDecPlacesWrong_Correct.
NumDecPlacesWrong
1
1.00
4
1 ATNumDecPlacesWrong_Correct.
Teacher uses displaydp
NumDecPlacesWrong
0.101
displaydp(101,3)
3
1 ATNumDecPlacesWrong_Correct.

## SigFigsStrict

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Basic tests
SigFigsStrict
3.141
null
SigFigsStrict
3.141
null
x^2
-1 STACKERROR_OPTION.
SigFigsStrict
3.141
null
-2
-1 STACKERROR_OPTION.
SigFigsStrict
3.141
null
0
-1 STACKERROR_OPTION.
SigFigsStrict
0.0010
null
1
0
SigFigsStrict
0.0010
null
2
1
SigFigsStrict
0.0010
null
3
0
SigFigsStrict
0.00100
null
2
0
SigFigsStrict
0.00100
null
3
1
SigFigsStrict
0.00100
null
4
0
SigFigsStrict
0.001
null
1
1
SigFigsStrict
0.001
null
2
0
SigFigsStrict
100
null
1
1
SigFigsStrict
100
null
2
0 ATSigFigsStrict_WithinRange.
SigFigsStrict
100
null
3
0 ATSigFigsStrict_WithinRange.
SigFigsStrict
100
null
4
0
SigFigsStrict
100.
null
1
0
SigFigsStrict
100.
null
2
0
SigFigsStrict
100.
null
3
1
SigFigsStrict
100.
null
4
0
SigFigsStrict
123.
null
1
0
SigFigsStrict
123.
null
2
0
SigFigsStrict
123.
null
3
1
SigFigsStrict
123.
null
4
0
SigFigsStrict
1.00e2
null
1
0
SigFigsStrict
1.00e2
null
2
0
SigFigsStrict
1.00e2
null
3
1
SigFigsStrict
1.00e2
null
4
0
SigFigsStrict
10.0
null
2
0
SigFigsStrict
10.0
null
3
1
SigFigsStrict
10.0
null
4
0
SigFigsStrict
0
null
1
1
SigFigsStrict
0
null
2
0
SigFigsStrict
0.0
null
1
1
SigFigsStrict
0.0
null
2
0
SigFigsStrict
.0
null
1
1
SigFigsStrict
.0
null
2
0
SigFigsStrict
.001030
null
4
1
SigFigsStrict
0.00
null
1
0
SigFigsStrict
0.00
null
2
1
SigFigsStrict
0.00
null
3
0
SigFigsStrict
25.00e1
null
1
0
SigFigsStrict
25.00e1
null
3
0
SigFigsStrict
25.00e1
null
4
1
SigFigsStrict
25.00e1
null
5
0
SigFigsStrict
15.1
15.1
3
1
SigFigsStrict
15.10
15.1
3
0
SigFigsStrict
15.100
15.1
3
0
Units are ignored
SigFigsStrict
9.81*m/s^2
null
3
1

## Units

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Units
1/0
1
2
Units
1
1/0
2
Units
1
1
1/0
Units
x-1)^2
(x-1)^2
2
Units
12.3*m*s^(-1)
3*m
[3,x]
Units
3*m*s^(-1)
3*m
[1,2,3]
Units
12.3*m*s^(-1)
{12.3*m*s^(-1)}
3
Units
x=12.3*m*s^(-1)
12.3*m*s^(-1)
3
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
12.3
12.3*m
3
Units
12
12.3*m
3
Units
1/2
12.3*m
3
Units
e^(1/2)
12.3*m
3
Units
9.81*m
12.3
3
Only units
Units
m/s
12.3*m/s
3
Units
m
12.3*m/s
3
Units
9.81+m/s
9.81*m/s
3
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
12.3*m/s
12.3*m/s
3
1 ATUnits_units_match.
Units
12.4*m/s
12.3*m/s
3
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
12.4*m/s
12.3*m/s
[3,2]
1 ATUnits_units_match.
Units
12.45*m/s
12.3*m/s
[3,2]
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATUnits_units_match.
Units
13.45*m/s
12.3*m/s
[3,2]
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
7.54E-5*(s*M)^-1
5.625E-5*s^-1
[3,2]
0 Your units are incompatible with those used by the teacher. ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
Units
7.54E-5*(s*M)^-1
stackunits(5.625E-5,1/s)
[3,2]
0 Your units are incompatible with those used by the teacher. ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
Units
12*m/s
12.3*m/s
3
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
-9.81*m/s^2
-9.81*m/s^2
3
1 ATUnits_units_match.
Units
-9.82*m/s^2
-9.815*m/s^2
3
1 ATUnits_units_match.
Units
-9.81*m/s^2
-9.815*m/s^2
3
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
-9.81*m*s^(-2)
-9.81*m/s^2
3
1 ATUnits_units_match.
Units
-9.82*m/s^2
-9.81*m/s^2
3
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
-9.81*m*s^(-2)
-9.81*m/s^2
3
1 ATUnits_units_match.
Units
-9.81*m/s/s
-9.81*m/s^2
3
1 ATUnits_units_match.
Units
-9.81*m/s
-9.81*m/s^2
3
0 Your units are incompatible with those used by the teacher. Please check your units carefully. ATUnits_incompatible_units. ATUnits_correct_numerical.
Units
-9.81*m/s
-9.81*m/s^2
3
0 Your units are incompatible with those used by the teacher. Please check your units carefully. ATUnits_incompatible_units. ATUnits_correct_numerical.
Units
(-9.81)*m/s^2
-9.81*m/s^2
3
1 ATUnits_units_match.
Units
520*amu
520*amu
3
1 ATNumSigFigs_WithinRange. ATUnits_units_match.
Units
520*amu
521*amu
3
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
(-9.81)
-9.81*m/s^2
3
Units
9.81*m/s
-9.81*m/s^2
3
0 Your answer has the wrong algebraic sign. Your units are incompatible with those used by the teacher. ATNumSigFigs_WrongSign. ATUnits_incompatible_units.
Units
8.81*m/s
-9.81*m/s^2
3
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
8.1*m/s
-9.81*m/s^2
3
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
m/4
0.25*m
3
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATUnits_units_match.
Student is too exact
Units
pi*s
3.14*s
3
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATUnits_units_match.
Units
sqrt(2)*m
1.41*m
3
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATUnits_units_match.
Different units
Units
25*g
0.025*kg
2
1 ATUnits_compatible_units kg.
Units
26*g
0.025*kg
2
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
100*g
10*kg
2
0 ATNumSigFigs_WithinRange. ATNumSigFigs_VeryInaccurate. ATUnits_compatible_units kg.
Units
0.025*g
0.025*kg
2
Units
1000*m
1*km
2
1 ATNumSigFigs_WithinRange. ATUnits_compatible_units m.
Units
1*Mg/10^6
1*N*s^2/(km)
1
1 ATUnits_compatible_units kg.
Units
1*Mg/10^6
1*kN*ns/(mm*Hz)
1
1 ATUnits_compatible_units kg.
Units
3.14*Mg/10^6
%pi*kN*ns/(mm*Hz)
3
1 ATUnits_compatible_units kg.
Units
3.141*Mg/10^6
%pi*kN*ns/(mm*Hz)
3
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATUnits_compatible_units kg.
Units
4.141*Mg/10^6
%pi*kN*ns/(mm*Hz)
3
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATNumSigFigs_VeryInaccurate. ATUnits_compatible_units kg.
Units
400*cc
0.4*l
2
1 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
Units
400*cm^3
0.4*l
2
1 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
Units
400*ml
0.4*l
2
1 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
Units
18*kJ
18000.0*J
2
1 ATUnits_compatible_units (kg*m^2)/s^2.
Units
18.1*kJ
18000.0*J
2
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATUnits_compatible_units (kg*m^2)/s^2.
Units
120*kWh
0.12*MWh
2
1 ATUnits_compatible_units (kg*m^2)/s^2.
Units
2.0*hh
720000*s
2
1 ATUnits_compatible_units s.
Units
723*kVA
0.723*MVA
3
1 ATUnits_compatible_units VA.
Edge case
Units
0*m/s
0*m/s
1
1 ATUnits_units_match.
Units
0.0*m/s
0*m/s
1
1 ATUnits_units_match.
Units
0*m/s
0.0*m/s
1
1 ATUnits_units_match.
Units
0.00*m/s
0.0*m/s
2
1 ATUnits_units_match.
Units
0.0*km/s
0.0*m/s
1
1 ATUnits_compatible_units m/s.
Units
0.0*m
0.0*m/s
1
0 Your units are incompatible with those used by the teacher. Please check your units carefully. ATUnits_incompatible_units. ATUnits_correct_numerical.
Units
0.0
0.0*m/s
1
Imperial
Units
7*in
7*in
1
1 ATUnits_units_match.
Units
6*in
0.5*ft
1
1 ATUnits_compatible_units in.
Units
2640*ft
0.5*mi
4
1 ATNumSigFigs_WithinRange. ATUnits_compatible_units in.
Units
2650*ft
0.5*mi
4
0 ATNumSigFigs_WithinRange. ATNumSigFigs_VeryInaccurate. ATUnits_compatible_units in.
TODO
Units !
142.8*C
415.9*K
4
-3 Your units are incompatible with those used by the teacher. ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
Units !
520*mamu
520*mamu
3

## UnitsStrict

Test
?
Student response
Opt
Mark
CAS errors
Feedback
Differences from the Units test only
UnitsStrict
25*g
0.025*kg
2
0 ATUnits_compatible_units kg.
UnitsStrict
1*Mg/10^6
1*N*s^2/(km)
1
0 ATUnits_compatible_units kg.
UnitsStrict
1*Mg/10^6
1*kN*ns/(mm*Hz)
1
0 ATUnits_compatible_units kg.
UnitsStrict
3.14*Mg/10^6
%pi*kN*ns/(mm*Hz)
3
0 ATUnits_compatible_units kg.
UnitsStrict
400*cc
0.4*l
2
0 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
UnitsStrict
400*cm^3
0.4*l
2
0 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
UnitsStrict
400*ml
0.4*l
2
0 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
UnitsStrict
400*mL
0.4*l
2
0 ATNumSigFigs_WithinRange. ATUnits_compatible_units m^3.
UnitsStrict
142.8*C
415.9*K
4
0 ATNumSigFigs_VeryInaccurate. ATUnits_incompatible_units.
We are not *that* strict!
UnitsStrict
-9.81*m/s/s
-9.81*m/s^2
3
1 ATUnits_units_match.
Edge case
UnitsStrict
0*m/s
0*m/s
1
1 ATUnits_units_match.
UnitsStrict
0.0*m/s
0*m/s
1
1 ATUnits_units_match.
UnitsStrict
0*m/s
0.0*m/s
1
1 ATUnits_units_match.
UnitsStrict
0.0*m/s
0.0*m/s
1
1 ATUnits_units_match.
UnitsStrict
0.0*km/s
0.0*m/s
1
0 ATUnits_compatible_units m/s.
UnitsStrict
0.0*m
0.0*m/s
1
0 ATUnits_incompatible_units. ATUnits_correct_numerical.
UnitsStrict
0.0
0.0*m/s
1
UnitsStrict
2.33e-15*kg
2.33e-15*kg
[3,2]
1 ATUnits_units_match.
UnitsStrict
7.03e-3*ng
7.03e-3*ng
[3,2]
1 ATUnits_units_match.
UnitsStrict
2.35e-6*ug
2.35e-6*ug
[3,2]
1 ATUnits_units_match.
UnitsStrict
9.83e-10*cg
9.83e-10*cg
[3,2]
1 ATUnits_units_match.
UnitsStrict
9.73e-21*Gg
9.73e-21*Gg
[3,2]
1 ATUnits_units_match.
UnitsStrict
7.19e-15*kg
7.19e-15*kg
[3,2]
1 ATUnits_units_match.
UnitsStrict
8.12e-12*g
8.12e-12*g
[3,2]
1 ATUnits_units_match.
UnitsStrict
9.34e-12*g
9.34e-12*g
[3,2]
1 ATUnits_units_match.
UnitsStrict
1.07e-21*Gg
1.07e-21*Gg
[3,2]
1 ATUnits_units_match.
UnitsStrict
1.91e-10*cg
1.91e-10*cg
[3,2]
1 ATUnits_units_match.
UnitsStrict
5.67e-18*Mg
5.67e-18*Mg
[3,2]
1 ATUnits_units_match.
UnitsStrict
2.04e-9*mg
2.04e-9*mg
[3,2]
1 ATUnits_units_match.
UnitsStrict
6.75e-6*ug
6.75e-6*ug
[3,2]
1 ATUnits_units_match.
UnitsStrict
6.58e-6*ug
6.58e-6*ug
[3,2]
1 ATUnits_units_match.
UnitsStrict
3.58e-9*mg
3.58e-9*mg
[3,2]
1 ATUnits_units_match.
UnitsStrict
9.99e-15*kg
9.99e-15*kg
[3,2]
1 ATUnits_units_match.
UnitsStrict
9.8e-9*mg
9.8e-9*mg
[3,2]
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATUnits_units_match.
UnitsStrict
9.80e-9*mg
9.8e-9*mg
[3,2]
1 ATUnits_units_match.
UnitsStrict
9.83e-9*mg
9.8e-9*mg
[3,2]
1 ATUnits_units_match.
UnitsStrict
9.78e-9*mg
9.8e-9*mg
[3,2]
1 ATUnits_units_match.
UnitsStrict
36*Kj/mol
36*Kj/mol
2
1 ATUnits_units_match.
UnitsStrict
-36*Kj/mol
-36*Kj/mol
2
1 ATUnits_units_match.
UnitsStrict
(-36)*Kj/mol
-36*Kj/mol
2
1 ATUnits_units_match.
UnitsStrict
(-36*Kj)/mol
-36*Kj/mol
2
1 ATUnits_units_match.
UnitsStrict
-(36*Kj)/mol
-36*Kj/mol
2
1 ATUnits_units_match.
UnitsStrict
-(36.2*Kj)/mol
-36.3*Kj/mol
2
0 Your answer contains the wrong number of significant digits. ATNumSigFigs_WrongDigits. ATUnits_units_match.

## UnitsRelative

Test
?
Student response
Opt
Mark
CAS errors
Feedback
UnitsRelative
12.3*m/s
12.3*m/s
0.01
1 ATUnits_units_match.
UnitsRelative
12*m/s
12.3*m/s
0.01
0 ATUnits_units_match.
UnitsRelative
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.15
1 ATUnits_compatible_units kg.
UnitsRelative
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.05
0 ATUnits_compatible_units kg.
Edge case
UnitsRelative
0*m/s
0*m/s
0.01
1 ATUnits_units_match.
UnitsRelative
0.0*m/s
0*m/s
0.01
1 ATUnits_units_match.
UnitsRelative
0*m/s
0.0*m/s
0.01
1 ATUnits_units_match.
UnitsRelative
0.0*m/s
0.0*m/s
0.01
1 ATUnits_units_match.
UnitsRelative
0.0*km/s
0.0*m/s
0.01
1 ATUnits_compatible_units m/s.
UnitsRelative
0.0*m
0.0*m/s
0.01
0 Your units are incompatible with those used by the teacher. Please check your units carefully. ATUnits_incompatible_units. ATUnits_correct_numerical.
UnitsRelative
0.0
0.0*m/s
0.01
UnitsRelative
0.0*kVA
0.0*kVA
0.002
1 ATUnits_units_match.

## UnitsStrictRelative

Test
?
Student response
Opt
Mark
CAS errors
Feedback
UnitsStrictRelative
12.3*m/s
12.3*m/s
0.01
1 ATUnits_units_match.
UnitsStrictRelative
12*m/s
12.3*m/s
0.01
0 ATUnits_units_match.
UnitsStrictRelative
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.15
0 ATUnits_compatible_units kg.
UnitsStrictRelative
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.05
0 ATUnits_compatible_units kg.
Edge case
UnitsStrictRelative
0*m/s
0*m/s
0.01
1 ATUnits_units_match.
UnitsStrictRelative
0.0*m/s
0*m/s
0.01
1 ATUnits_units_match.
UnitsStrictRelative
0*m/s
0.0*m/s
0.01
1 ATUnits_units_match.
UnitsStrictRelative
0.0*m/s
0.0*m/s
0.01
1 ATUnits_units_match.
UnitsStrictRelative
0.0*km/s
0.0*m/s
0.01
0 ATUnits_compatible_units m/s.
UnitsStrictRelative
0.0*m
0.0*m/s
0.01
0 ATUnits_incompatible_units. ATUnits_correct_numerical.
UnitsStrictRelative
0.0
0.0*m/s
0.01
UnitsStrictRelative
0*J
0.0*J
0.01
1 ATUnits_units_match.

## UnitsAbsolute

Test
?
Student response
Opt
Mark
CAS errors
Feedback
UnitsAbsolute
-123000*J
-123*kJ
5*J
UnitsAbsolute
12.3*m/s
12.3*m/s
0.01
1 ATUnits_units_match.
UnitsAbsolute
12*m/s
12.3*m/s
0.01
0 ATUnits_units_match.
UnitsAbsolute
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.15
1 ATUnits_compatible_units kg.
The following illustrates that we convert to base units to compare.
UnitsAbsolute
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.1
1 ATUnits_compatible_units kg.
UnitsAbsolute
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.09
0 ATUnits_compatible_units kg.
Units in the options
UnitsAbsolute
-123000*J
-123*kJ
5*kJ
1 ATUnits_compatible_units (kg*m^2)/s^2.
UnitsAbsolute
-123006*J
-123*kJ
5*kJ
1 ATUnits_compatible_units (kg*m^2)/s^2.
UnitsAbsolute
-129006*J
-123*kJ
5*kJ
0 ATUnits_compatible_units (kg*m^2)/s^2.
UnitsAbsolute
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.1*kN*ns/(mm*H
z)
1 ATUnits_compatible_units kg.
UnitsAbsolute
1.1*Mg/10^6
1.2*kN*ns/(mm*Hz)
0.09*kN*ns/(mm*
Hz)
0 ATUnits_compatible_units kg.
Edge case
UnitsAbsolute
0*m/s
0*m/s
0.01
1 ATUnits_units_match.
UnitsAbsolute
0.0*m/s
0*m/s
0.01
1 ATUnits_units_match.
UnitsAbsolute
0*m/s
0.0*m/s
0.01
1 ATUnits_units_match.
UnitsAbsolute
0.0*m/s
0.0*m/s
0.01
1 ATUnits_units_match.
UnitsAbsolute
0.0*km/s
0.0*m/s
0.01
1 ATUnits_compatible_units m/s.
UnitsAbsolute
0.0*m
0.0*m/s
0.01
0 Your units are incompatible with those used by the teacher. Please check your units carefully. ATUnits_incompatible_units. ATUnits_correct_numerical.
UnitsAbsolute
0.0
0.0*m/s
0.01
UnitsAbsolute
1.0*m/s
m/s
0.01
1 ATUnits_units_match.
UnitsAbsolute
15/pi*kN/mm^2
15/pi*kN/mm^2
0.01
1 ATUnits_units_match.
UnitsAbsolute
(15*kN)/(pi*mm^2)
(15*kN)/(pi*mm^2)
0.01
1 ATUnits_units_match.
UnitsAbsolute
(15/pi)*(kN/mm^2)
(15/pi)*(kN/mm^2)
0.01
1 ATUnits_units_match.
UnitsAbsolute
(600*N)/(%pi*mm^2)
(600*N)/(%pi*mm^2)
0.01
1 ATUnits_units_match.
UnitsAbsolute
(600/pi)*kN/m^2
(600/pi)*kN/m^2
0.01
1 ATUnits_units_match.
UnitsAbsolute
(600/pi)*kN/mm^2
(600/pi)*kN/mm^2
0.01
1 ATUnits_units_match.