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Equiv: Answer test results

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

Equiv

Test
?
Student response
Teacher answer
Opt
Mark
Answer note
Equiv 
x
 
[x^2=4,x=2 or x=-2]
 -1 ATEquiv_SA_not_list.
The first argument to the Equiv answer test should be a list, but the test failed. Please contact your teacher.
Equiv 
[x^2=4,x=2 or x=-2]
 
x
 -1 ATEquiv_SB_not_list.
The second argument to the Equiv answer test should be a list, but the test failed. Please contact your teacher.
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 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &x^2=4& \\cr \\color{green}{\\Leftrightarrow}&x=2\\,{\\mbox{ or }}\\, x=-2& \\cr \\end{array}\\]
Equiv 
[x^2=4,x=#pm#2,x=2 and x=-2]
 
[x^2=4,x=2 or x=-2]
 0 (EMPTYCHAR, EQUIVCHAR,ANDOR)
\\[\\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}\\]
Equiv 
[x^2=4,x=2]
 
[x^2=4,x=2 or x=-2]
 0 (EMPTYCHAR,IMPLIEDCHAR)
\\[\\begin{array}{lll} &x^2=4& \\cr \\color{red}{\\Leftarrow}&x=2& \\cr \\end{array}\\]
Equiv 
[x^2=4,x=2]
 
[x^2=4,x=2]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&x^2=4& \\cr \\color{green}{\\Leftrightarrow}&x=2& \\cr \\end{array}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2=4,x= #pm#2, x=2 or x=-2]
 
[x^2=4,x=2 or x=-2]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2-6*x+9=0,x=3]
 
[x^2-6*x+9=0,x=3]
 1 (EMPTYCHAR,SAMEROOTS)
\\[\\begin{array}{lll} &x^2-6\\cdot x+9=0& \\cr \\color{green}{\\mbox{(Same roots)}}&x=3& \\cr \\end{array}\\]
Equiv 
[]
 
[]
 1 (EMPTYCHAR)
\\[\\begin{array}{lll} &\\left[ \\right] & \\cr \\end{array}\\]
Equiv 
[x^2=-1]
 
[]
 1 (EMPTYCHAR)
\\[\\begin{array}{lll} &x^2=-1& \\cr \\end{array}\\]
Equiv 
[x=x,all]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &x=x& \\cr \\color{green}{\\Leftrightarrow}&\\mathbb{R}& \\cr \\end{array}\\]
Equiv 
[x=x,true]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &x=x& \\cr \\color{green}{\\Leftrightarrow}&\\mathbf{True}& \\cr \\end{array}\\]
Equiv 
[x=x,false]
 
[]
 0 (EMPTYCHAR,QMCHAR)
\\[\\begin{array}{lll} &x=x& \\cr \\color{red}{?}&\\mathbf{False}& \\cr \\end{array}\\]
Equiv 
[1=1,all]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &1=1& \\cr \\color{green}{\\Leftrightarrow}&\\mathbb{R}& \\cr \\end{array}\\]
Equiv 
[1=1,true]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &1=1& \\cr \\color{green}{\\Leftrightarrow}&\\mathbf{True}& \\cr \\end{array}\\]
Equiv 
[0=0,all]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &0=0& \\cr \\color{green}{\\Leftrightarrow}&\\mathbb{R}& \\cr \\end{array}\\]
Equiv 
[0=0,true]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &0=0& \\cr \\color{green}{\\Leftrightarrow}&\\mathbf{True}& \\cr \\end{array}\\]
Equiv 
[1=2,false]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &1=2& \\cr \\color{green}{\\Leftrightarrow}&\\mathbf{False}& \\cr \\end{array}\\]
Equiv 
[1=2,none]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &1=2& \\cr \\color{green}{\\Leftrightarrow}&\\emptyset& \\cr \\end{array}\\]
Equiv 
[1=2,{}]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &1=2& \\cr \\color{green}{\\Leftrightarrow}&\\left \\{ \\right \\}& \\cr \\end{array}\\]
Equiv 
[1=2,[]]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &1=2& \\cr \\color{green}{\\Leftrightarrow}&\\left[ \\right] & \\cr \\end{array}\\]
Equiv 
[x=1,X=1]
 
[]
 0 (EMPTYCHAR,QMCHAR)
\\[\\begin{array}{lll} &x=1& \\cr \\color{red}{?}&X=1& \\cr \\end{array}\\]
Equiv 
[1/(x^2+1)=1/((x+%i)*(x-%i)),t
rue]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[2^2,stackeq(4)]
 
[]
 1 (EMPTYCHAR, CHECKMARK)
\\[\\begin{array}{lll} &2^2& \\cr \\color{green}{\\checkmark}&=4& \\cr \\end{array}\\]
Equiv 
[2^2,stackeq(3)]
 
[]
 0 (EMPTYCHAR,IMPLIESCHAR)
\\[\\begin{array}{lll} &2^2& \\cr \\color{red}{\\Rightarrow}&=3& \\cr \\end{array}\\]
Equiv 
[2^2,4]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &2^2& \\cr \\color{green}{\\Leftrightarrow}&4& \\cr \\end{array}\\]
Equiv 
[2^2,3]
 
[]
 0 (EMPTYCHAR,IMPLIESCHAR)
\\[\\begin{array}{lll} &2^2& \\cr \\color{red}{\\Rightarrow}&3& \\cr \\end{array}\\]
Equiv 
[lg(64,4),lg(4^3,4),3*lg(4,4),
3]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[lg(64,4),stackeq(lg(4^3,4)),s
tackeq(3*lg(4,4)),stackeq(3)]
 
[]
 1 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\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}\\]
Equiv 
[x=1 or x=2,x=1 or 2]
 
[]
 0 (EMPTYCHAR,MISSINGVAR)
\\[\\begin{array}{lll} &x=1\\,{\\mbox{ or }}\\, x=2& \\cr \\color{red}{\\mbox{Missing assignments}}&x=1\\,{\\mbox{ or }}\\, 2& \\cr \\end{array}\\]
Equiv 
[x=1 or x=2,x=1 and x=2]
 
[]
 0 (EMPTYCHAR,ANDOR)
\\[\\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}\\]
Equiv 
[x=1 and y=2,x=1 or y=2]
 
[]
 0 (EMPTYCHAR,ANDOR)
\\[\\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}\\]
Equiv 
[a=b,a^2=b^2]
 
[]
 0 (EMPTYCHAR,IMPLIESCHAR)
\\[\\begin{array}{lll} &a=b& \\cr \\color{red}{\\Rightarrow}&a^2=b^2& \\cr \\end{array}\\]
Equiv 
[a=b,sqrt(a)=sqrt(b)]
 
[]
 0 (EMPTYCHAR,IMPLIEDCHAR)
\\[\\begin{array}{lll} &a=b& \\cr \\color{red}{\\Leftarrow}&\\sqrt{a}=\\sqrt{b}& \\cr \\end{array}\\]
Equiv 
[a^2=b^2,a=b]
 
[]
 0 (EMPTYCHAR,IMPLIEDCHAR)
\\[\\begin{array}{lll} &a^2=b^2& \\cr \\color{red}{\\Leftarrow}&a=b& \\cr \\end{array}\\]
Equiv 
[a^2=b^2,a=b or a=-b]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &a^2=b^2& \\cr \\color{green}{\\Leftrightarrow}&a=b\\,{\\mbox{ or }}\\, a=-b& \\cr \\end{array}\\]
Equiv 
[a^2=b^2,a= #pm#b,a= b or a=-b
]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,SAMEROOTS)
\\[\\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}\\]
Equiv 
[a=b,abs(a)=abs(b),a=b]
 
[]
 0 (EMPTYCHAR,IMPLIESCHAR,IMPLIEDCHAR)
\\[\\begin{array}{lll} &a=b& \\cr \\color{red}{\\Rightarrow}&\\left| a\\right| =\\left| b\\right| & \\cr \\color{red}{\\Leftarrow}&a=b& \\cr \\end{array}\\]
Equiv 
[abs(a)=abs(b),a=b or a=-b]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &\\left| a\\right| =\\left| b\\right| & \\cr \\color{green}{\\Leftrightarrow}&a=b\\,{\\mbox{ or }}\\, a=-b& \\cr \\end{array}\\]
Equiv 
[abs(a)=abs(b),a^2=b^2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\begin{array}{lll} &\\left| a\\right| =\\left| b\\right| & \\cr \\color{green}{\\Leftrightarrow}&a^2=b^2& \\cr \\end{array}\\]
Equiv 
[x^3=8,x=2]
 
[]
 0 (EMPTYCHAR,IMPLIEDCHAR)
\\[\\begin{array}{lll} &x^3=8& \\cr \\color{red}{\\Leftarrow}&x=2& \\cr \\end{array}\\]
Equiv 
[x^3=8,x=2]
 
[]
 
[assumereal]
 1 (ASSUMEREALVARS, EQUIVCHARREAL)
\\[\\begin{array}{lll}\\color{blue}{(\\mathbb{R})}&x^3=8& \\cr \\color{green}{\\Leftrightarrow}\\, \\color{blue}{(\\mathbb{R})}&x=2& \\cr \\end{array}\\]
Equiv 
[abs(x-1/2)+abs(x+1/2)=2,abs(x
)=1]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[a^2=9 and a>0,a=3]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[T=2*pi*sqrt(L/g),T^2=4*pi^2*L
/g,g=4*pi^2*L/T^2]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[a=b,a^2=b^2]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&a=b& \\cr \\color{green}{\\Leftrightarrow}&a^2=b^2& \\cr \\end{array}\\]
Equiv 
[a=b,sqrt(a)=sqrt(b)]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&a=b& \\cr \\color{green}{\\Leftrightarrow}&\\sqrt{a}=\\sqrt{b}& \\cr \\end{array}\\]
Equiv 
[a^2=b^2,a=b]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&a^2=b^2& \\cr \\color{green}{\\Leftrightarrow}&a=b& \\cr \\end{array}\\]
Equiv 
[a^2=b^2,a=b or a=-b]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[a=b,abs(a)=abs(b)]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&a=b& \\cr \\color{green}{\\Leftrightarrow}&\\left| a\\right| =\\left| b\\right| & \\cr \\end{array}\\]
Equiv 
[abs(a)=abs(b),a=b]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&\\left| a\\right| =\\left| b\\right| & \\cr \\color{green}{\\Leftrightarrow}&a=b& \\cr \\end{array}\\]
Equiv 
[abs(a)=abs(b),a=-b]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&\\left| a\\right| =\\left| b\\right| & \\cr \\color{green}{\\Leftrightarrow}&a=-b& \\cr \\end{array}\\]
Equiv 
[abs(a)=abs(b),a=b or a=-b]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x=abs(-2),x=2]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&x=\\left| -2\\right| & \\cr \\color{green}{\\Leftrightarrow}&x=2& \\cr \\end{array}\\]
Equiv 
[abs(a)=abs(b),a^2=b^2]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2=9,x=#pm#3,x=3 or x=-3,x=3
]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2=9,x=3]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&x^2=9& \\cr \\color{green}{\\Leftrightarrow}&x=3& \\cr \\end{array}\\]
Equiv 
[x^2=2,x=#pm#sqrt(2),x=sqrt(2)
 or x=-sqrt(2)]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2=2,x=sqrt(2)]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\begin{array}{lll}\\color{blue}{\\mbox{Assume +ve vars}}&x^2=2& \\cr \\color{green}{\\Leftrightarrow}&x=\\sqrt{2}& \\cr \\end{array}\\]
Equiv 
[x^2 = a^2-b,x = sqrt(a^2-b)]
 
[]
 
[assumepos]
 1 (ASSUMEPOSVARS, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[2*(x-3) = 4*x-3*(x+2),2*x-6=x
-6,x=0]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[2*(x-3) = 5*x-3*(x+2),2*x-6=2
*x-6,0=0,all]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[2*(x-3) = 5*x-3*(x+1),2*x-6=2
*x-3,0=3,{}]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[a^2=b^2,a^2-b^2=0,(a-b)*(a+b)
=0,a=b or a=-b]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2=2,x^2-2=0,(x-sqrt(2))*(x+
sqrt(2))=0,x=sqrt(2) or x=-sqr
t(2)]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2=2,x=#pm#sqrt(2),x=sqrt(2)
 or x=-sqrt(2)]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2-6*x=-9,(x-3)^2=0,x-3=0,x=
3]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR,SAMEROOTS, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^3-1=0,(x-1)*(x^2+x+1)=0,x=1
]
 
[]
 0 (EMPTYCHAR, EQUIVCHAR,IMPLIEDCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (ASSUMEPOSVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x*(x-1)*(x-2)=0,x*(x-1)=0,x*(
x-1)*(x-2)=0,x*(x^2-2)=0]
 
[]
 0 (EMPTYCHAR,IMPLIEDCHAR,IMPLIESCHAR,QMCHAR)
\\[\\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}\\]
Equiv 
[x^2-6*x=-9,x=3]
 
[]
 1 (EMPTYCHAR,SAMEROOTS)
\\[\\begin{array}{lll} &x^2-6\\cdot x=-9& \\cr \\color{green}{\\mbox{(Same roots)}}&x=3& \\cr \\end{array}\\]
Equiv 
[x=1 nounor x=-2 nounor x=1,x^
3-3*x=-2,x=1 nounor x=-2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR,SAMEROOTS)
\\[\\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}\\]
Equiv 
[9*x^3-24*x^2+13*x=2,x=1/3 nou
nor x=2]
 
[]
 1 (EMPTYCHAR,SAMEROOTS)
\\[\\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}\\]
Equiv 
[(x-2)^43*(x+1/3)^60=0,(3*x+1)
^4*(x-2)^2=0,x=-1/3 nounor x=2
]
 
[]
 1 (EMPTYCHAR,SAMEROOTS,SAMEROOTS)
\\[\\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}\\]
Equiv 
[2^x=4,x*log(2)=log(4),x=log(2
^2)/log(2),x=2*log(2)/log(2),x
=2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^log(y),stackeq(e^(log(x)*lo
g(y))),stackeq(e^(log(y)*log(x
))),stackeq(y^log(x))]
 
[]
 1 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,EQUIVLOG, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[a=logbase(9,3),3^a=9,3^a=3^2,
a=2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR,QMCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^3-1=0,(x-1)*(x^2+x+1)=0,x=1
]
 
[]
 
[assumereal]
 1 (ASSUMEREALVARS, EQUIVCHAR, EQUIVCHARREAL)
\\[\\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}\\]
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 (ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHARREAL, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[6*x-12=3*(x-2),6*x-12+3*(x-2)
=0,9*x-18=0,x=2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,SAMEROOTS)
\\[\\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}\\]
Equiv 
[(x+3)*(2-x)=4,x+3=4 or (2-x)=
4,x=1 or x=-2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR,IMPLIEDCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[(x^2-4)/(x-2)=0,(x-2)*(x+2)/(
x-2)=0,x+2=0,x=-2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[(x^2-4)/(x-2)=0,(x^2-4)=0,(x-
2)*(x+2)=0,x=-2 or x=2]
 
[]
 0 (EMPTYCHAR,IMPLIESCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,IMPLIEDCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[a*x^2+b*x+c=0,a=0 nounand b=0
 nounand c=0,a*x^2+b*x+c=0]
 
[]
 1 (EMPTYCHAR,EQUATECOEFFLOSS(x),EQUATECOEFFGAIN(x))
\\[\\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}\\]
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 (EMPTYCHAR,EQUATECOEFFLOSS(x),EQUATECOEFFGAIN(x))
\\[\\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}\\]
Equiv 
[(x-1)*(x+4), stackeq(x^2-x+4*
x-4),stackeq(x^2+3*x-4)]
 
[]
 1 (EMPTYCHAR, CHECKMARK, CHECKMARK)
\\[\\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}\\]
Equiv 
[(x-1)*(x+4), stackeq(x^2-x+4*
x-4),stackeq(x^2+3*x-4)]
 
[]
 1 (EMPTYCHAR, CHECKMARK, CHECKMARK)
\\[\\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}\\]
Equiv 
[x^2-2,stackeq((x-sqrt(2))*(x+
sqrt(2)))]
 
[]
 1 (EMPTYCHAR, CHECKMARK)
\\[\\begin{array}{lll} &x^2-2& \\cr \\color{green}{\\checkmark}&=\\left(x-\\sqrt{2}\\right)\\cdot \\left(x+\\sqrt{2}\\right)& \\cr \\end{array}\\]
Equiv 
[x^2+4,stackeq((x-2*i)*(x+2*i)
)]
 
[]
 1 (EMPTYCHAR, CHECKMARK)
\\[\\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}\\]
Equiv 
[x^2+2*a*x,x^2+2*a*x+a^2-a^2,(
x+a)^2-a^2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2+2*a*x,stackeq(x^2+2*a*x+a
^2-a^2),stackeq((x+a)^2-a^2)]
 
[]
 1 (EMPTYCHAR, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\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}\\]
Equiv 
[abs(x-1/2)+abs(x+1/2)-2,stack
eq(abs(x)-1)]
 
[]
 0 (EMPTYCHAR,QMCHAR)
\\[\\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}\\]
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 (EMPTYCHAR,QMCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[1/(x^2+1)=1/((x+%i)*(x-%i)), 
stackeq(1/(2*%i)*(1/(x-%i)-1/(
x+%i)))]
 
[]
 1 (CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\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}\\]
Equiv 
[binomial(n,k)+binomial(n,k-1)
,stackeq(n!/((k-1)!*(n-k+1)!)+
n!/(k!*(n-k)!)),stackeq(n!*k/(
k!*(n-k+1)!)+n!*(n-k+1)/(k!*(n
-k+1)!)),stackeq(n!*k/(k!*(n-k
+1)!)+n!/(k!*(n-k)!)),stackeq(
((n-k+1)*n!+k*n!)/(k!*(n-k+1)!
)),stackeq(((n+1)*n!)/(k!*(n-k
+1)!))]
 
[]
 1 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\begin{array}{lll} &{{n}\\choose{k}}+{{n}\\choose{k-1}}& \\cr \\color{green}{\\checkmark}&=\\frac{n!}{\\left(k-1\\right)!\\cdot \\left(n-k+1\\right)!}+\\frac{n!}{k!\\cdot \\left(n-k\\right)!}& \\cr \\color{green}{\\checkmark}&=\\frac{n!\\cdot k}{k!\\cdot \\left(n-k+1\\right)!}+\\frac{n!\\cdot \\left(n-k+1\\right)}{k!\\cdot \\left(n-k+1\\right)!}& \\cr \\color{green}{\\checkmark}&=\\frac{n!\\cdot k}{k!\\cdot \\left(n-k+1\\right)!}+\\frac{n!}{k!\\cdot \\left(n-k\\right)!}& \\cr \\color{green}{\\checkmark}&=\\frac{\\left(n-k+1\\right)\\cdot n!+k\\cdot n!}{k!\\cdot \\left(n-k+1\\right)!}& \\cr \\color{green}{\\checkmark}&=\\frac{\\left(n+1\\right)\\cdot n!}{k!\\cdot \\left(n-k+1\\right)!}& \\cr \\end{array}\\]
Equiv 
[(x-1)^2=(x-1)*(x-1), stackeq(
x^2-2*x+1)]
 
[]
 1 (CHECKMARK, CHECKMARK)
\\[\\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}\\]
Equiv 
[(x-1)^2=(x-1)*(x-1), stackeq(
x^2-2*x+2)]
 
[]
 0 (CHECKMARK,QMCHAR)
\\[\\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}\\]
Equiv 
[(x-2)^2=(x-1)*(x-1), stackeq(
x^2-2*x+1)]
 
[]
 0 (QMCHAR, CHECKMARK)
\\[\\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}\\]
Equiv 
[4^((n+1)+1)-1= 4*4^(n+1)-1,st
ackeq(4*(4^(n+1)-1)+3)]
 
[]
 1 (CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[a^2=b and x=1, b=a^2 and x=1]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (ASSUMEREALVARS, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, CHECKMARK, EMPTYCHAR, EMPTYCHAR, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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&gt
;=3/2 or x <=-2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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&gt
;=3/2 or x <=-2]
 
[]
 1 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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&gt
;=3/2 or x <=2]
 
[]
 0 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR,QMCHAR,QMCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EMPTYCHAR, EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, EMPTYCHAR, EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, CHECKMARK, EMPTYCHAR, EMPTYCHAR, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[x^2+1,x^3/3+x,x^2+1,x^3/3+x+c
]
 
[]
 
[calculus]
 1 (EMPTYCHAR,INTCHAR(x),DIFFCHAR(x),INTCHAR(x))
\\[\\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}\\]
Equiv 
[3*x^(3/2)-2/x,(9*sqrt(x))/2+2
/x^2,3*x^(3/2)-2/x+c]
 
[]
 
[calculus]
 1 (EMPTYCHAR,DIFFCHAR(x),INTCHAR(x))
\\[\\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}\\]
Equiv 
[x^2+1,stackeq(x^3/3+x),stacke
q(x^2+1),stackeq(x^3/3+x+c)]
 
[]
 
[calculus]
 0 (EMPTYCHAR,QMCHAR,QMCHAR,QMCHAR)
\\[\\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}\\]
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 (EMPTYCHAR, CHECKMARK, CHECKMARK)
\\[\\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}\\]
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 (EMPTYCHAR,DIFFCHAR(x), EQUIVCHAR, EQUIVCHAR)
\\[\\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}\\]
Equiv 
[nounint(s^2+1,s),stackeq(s^3/
3+s+c)]
 
[]
 
[calculus]
 1 (EMPTYCHAR,INTCHAR(s))
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR,PLUSC)
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR,INTCHAR(x))
\\[\\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}\\]
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 (EMPTYCHAR, EQUIVCHAR, EQUIVCHAR,INTCHAR(x),INTCHAR(x))
\\[\\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}\\]
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