How can I create some versions of exercises and get the LaTeX file?

Question

Sometimes, I want to create some versions of exercises with the same content. For example, solve the equations:

1. $x^2 - 5x - 6 =0$;
2. $x^2 + 5x - 6 =0$;
3. $2x^2 + 5x - 7 =0$.

Then I solve each equation in my document.

How can I create some versions of exercises and get a LaTeX file?

Answer 1

(edit: added simplification if needed)

This output

is generated by this code (note: code is written in CODE cell (not INPUT) cell). To copy please copy to code cell, else it will get mis-formatted).

Here is link to notebook also.

To use, add your quadratic equation to the list eqs below and run the whole code. It will create file A.tex in same folder as notebook. Compile the A.tex file using lualatex or pdflatex or any Latex compiler to obtain the PDF showing the step by step solution.

eqs={x^2-5*x-6==0,x^2+5*x-6==0,2*x^2+5*x-7==0,Sqrt[2] x^2+x-(Sqrt[2]+1)==0}
toX[e_]:=ToString[TeXForm[e]];
formatA[eqIn_Equal,x_]:=Module[{eq=eqIn,s,lis,a,b,c,discriminant},
    eq=First@eq-Last@eq;
    lis=CoefficientList[eq,x]; 
    Print["lis=",lis];
    If[Length[lis]!=3,Abort];
    c=lis[[1]];b=lis[[2]];a=lis[[3]];
    discriminant=b^2-4*a*c; 
    If[LeafCount[Simplify@discriminant]<LeafCount[discriminant],
        s="\\[\n\\begin{array}[c]{cccccc}\n"<>
          "\\Delta &= b^2 - 4 a c &=\\left("<> toX[b]<>"\\right)^2- 4\\left("<>toX[a]<>"\\right)\\left("<>toX[c]<>"\\right)"<>
          "&="<>toX[discriminant]<>"&="<>toX[Simplify@discriminant]<>
          "\n\\end{array}\n\\]\n";
        discriminant=Simplify@discriminant
    ,
        s="\\[\n\\begin{array}[c]{cccc}\n"<>
          "\\Delta &= b^2 - 4 a c &=\\left("<> toX[b]<>"\\right)^2- 4\\left("<>toX[a]<>"\\right)\\left("<>toX[c]<>"\\right)"<>
          "&="<>toX[discriminant]<>
          "\n\\end{array}\n\\]\n";
   ];
   {s,a,b,c,discriminant}
];

formatB[a_,b_,c_,x_,disc_]:=Module[{s,sol},
      sol=(-b+Sqrt[disc])/(2*a);
      If[LeafCount[FullSimplify[sol]]<LeafCount[sol],
         s="\\[\n\\begin{array}[c]{ccccc}\n"<>
             toX[x]<>"_1 &= \\frac{-b + \\sqrt{\\Delta}}{2a}"<>
             "&= \\frac{-\\left("<>toX[b]<>"\\right) + \\sqrt{"<>toX[disc]<>"}}{2\\left("<>toX[a]<>"\\right)} "<>
             "&= "<>toX[sol]<>" &="<>toX[FullSimplify[sol]]<>"\n\\end{array}\n\\]\n";
         ,
         s="\\[\n\\begin{array}[c]{cccc}\n"<>
             toX[x]<>"_1 &= \\frac{-b + \\sqrt{\\Delta}}{2a}"<>
             "&= \\frac{-\\left("<>toX[b]<>"\\right) + \\sqrt{"<>toX[disc]<>"}}{2\\left("<>toX[a]<>"\\right)} "<>
             "&= "<>toX[sol]<>"\n\\end{array}\n\\]\n";
      ];
      s=s<>"And\n";
      
      sol=(-b-Sqrt[disc])/(2*a);
      If[LeafCount[FullSimplify[sol]]<LeafCount[sol],
         s=s<>"\\[\n\\begin{array}[c]{ccccc}\n"<>
             toX[x]<>"_2 &= \\frac{-b - \\sqrt{\\Delta}}{2a}"<>
             "&= \\frac{-\\left("<>toX[b]<>"\\right) - \\sqrt{"<>toX[disc]<>"}}{2\\left("<>toX[a]<>"\\right)} "<>
             "&= "<>toX[sol]<>" &="<>toX[FullSimplify[sol]]<>"\n\\end{array}\n\\]\n";
         ,
         s=s<>"\\[\n\\begin{array}[c]{cccc}\n"<>
             toX[x]<>"_2 &= \\frac{-b - \\sqrt{\\Delta}}{2a}"<>
             "&= \\frac{-\\left("<>toX[b]<>"\\right) - \\sqrt{"<>toX[disc]<>"}}{2\\left("<>toX[a]<>"\\right)} "<>
             "&= "<>toX[sol]<>"\n\\end{array}\n\\]\n";
      ];
        
      s
];

SetDirectory[NotebookDirectory[]]
fileName=FileNameJoin[{Directory[],"A.tex"}]
If[FileExistsQ[fileName],DeleteFile[fileName]];
file=OpenWrite[fileName,PageWidth->Infinity];
WriteString[file,"\\documentclass[12pt,a4paper]{article}\n"<>
       "\\usepackage[margin=1in]{geometry}\n"<>
       "\\usepackage{amsmath}\n"<>
        "\\usepackage{mathtools}\n"<>
        "\\begin{document}\n"
];

WriteString[file,"Solve the following equations\n\\begin{enumerate}\n"];
Do[WriteString[file,"\\item $"<>toX[eqs[[counter]]]<>"$\n"],{counter,1,Length@eqs}];
WriteString[file,"\\end{enumerate}\n"];
WriteString[file,"\\begin{enumerate}\n"];

Do[currentEq=eqs[[counter]];
    {s,a,b,c,discriminant}=formatA[currentEq,x];
    WriteString[file,"\\item For the equation $"<>toX[currentEq]<>"$ we have\n"<>s<>
        "The given equation has two solutions\n"<>formatB[a,b,c,x,discriminant]
    ],
    {counter,1,Length@eqs}
];

WriteString[file,"\\end{enumerate}\n\\end{document}\n"];
Close[file]

And this is the latex file

\documentclass[12pt,a4paper]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
\usepackage{mathtools}
\begin{document}
Solve the following equations
\begin{enumerate}
\item $x^2-5 x-6=0$
\item $x^2+5 x-6=0$
\item $2 x^2+5 x-7=0$
\item $\sqrt{2} x^2+x-\sqrt{2}-1=0$
\end{enumerate}
\begin{enumerate}
\item For the equation $x^2-5 x-6=0$ we have
\[
\begin{array}[c]{cccc}
\Delta &= b^2 - 4 a c &=\left(-5\right)^2- 4\left(1\right)\left(-6\right)&=49
\end{array}
\]
The given equation has two solutions
\[
\begin{array}[c]{cccc}
x_1 &= \frac{-b + \sqrt{\Delta}}{2a}&= \frac{-\left(-5\right) + \sqrt{49}}{2\left(1\right)} &= 6
\end{array}
\]
And
\[
\begin{array}[c]{cccc}
x_2 &= \frac{-b - \sqrt{\Delta}}{2a}&= \frac{-\left(-5\right) - \sqrt{49}}{2\left(1\right)} &= -1
\end{array}
\]
\item For the equation $x^2+5 x-6=0$ we have
\[
\begin{array}[c]{cccc}
\Delta &= b^2 - 4 a c &=\left(5\right)^2- 4\left(1\right)\left(-6\right)&=49
\end{array}
\]
The given equation has two solutions
\[
\begin{array}[c]{cccc}
x_1 &= \frac{-b + \sqrt{\Delta}}{2a}&= \frac{-\left(5\right) + \sqrt{49}}{2\left(1\right)} &= 1
\end{array}
\]
And
\[
\begin{array}[c]{cccc}
x_2 &= \frac{-b - \sqrt{\Delta}}{2a}&= \frac{-\left(5\right) - \sqrt{49}}{2\left(1\right)} &= -6
\end{array}
\]
\item For the equation $2 x^2+5 x-7=0$ we have
\[
\begin{array}[c]{cccc}
\Delta &= b^2 - 4 a c &=\left(5\right)^2- 4\left(2\right)\left(-7\right)&=81
\end{array}
\]
The given equation has two solutions
\[
\begin{array}[c]{cccc}
x_1 &= \frac{-b + \sqrt{\Delta}}{2a}&= \frac{-\left(5\right) + \sqrt{81}}{2\left(2\right)} &= 1
\end{array}
\]
And
\[
\begin{array}[c]{cccc}
x_2 &= \frac{-b - \sqrt{\Delta}}{2a}&= \frac{-\left(5\right) - \sqrt{81}}{2\left(2\right)} &= -\frac{7}{2}
\end{array}
\]
\item For the equation $\sqrt{2} x^2+x-\sqrt{2}-1=0$ we have
\[
\begin{array}[c]{cccccc}
\Delta &= b^2 - 4 a c &=\left(1\right)^2- 4\left(\sqrt{2}\right)\left(-1-\sqrt{2}\right)&=1-4 \sqrt{2} \left(-1-\sqrt{2}\right)&=9+4 \sqrt{2}
\end{array}
\]
The given equation has two solutions
\[
\begin{array}[c]{ccccc}
x_1 &= \frac{-b + \sqrt{\Delta}}{2a}&= \frac{-\left(1\right) + \sqrt{9+4 \sqrt{2}}}{2\left(\sqrt{2}\right)} &= \frac{\sqrt{9+4 \sqrt{2}}-1}{2 \sqrt{2}} &=1
\end{array}
\]
And
\[
\begin{array}[c]{ccccc}
x_2 &= \frac{-b - \sqrt{\Delta}}{2a}&= \frac{-\left(1\right) - \sqrt{9+4 \sqrt{2}}}{2\left(\sqrt{2}\right)} &= \frac{-1-\sqrt{9+4 \sqrt{2}}}{2 \sqrt{2}} &=-1-\frac{1}{\sqrt{2}}
\end{array}
\]
\end{enumerate}
\end{document}

Answer 2

Question Challenge: Non-Mathematica Answer.

I hope this is not against the rules of the site, but I suggest that the best tool for this job is not Mathematica.

If you can compile in LuaLaTeX, the luacas package allows you to do exactly this sort of computation / replacement straight in the .tex file, with very simple syntax.

And in case you are interested, I am working on a similar system with random generation of quadratic equations, which is now available on GitHub.