Problem:
Find the roots of the following polynomial: $15 x^2-\frac{x}{7}-9$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Solve for }x: \\
 15 x^2-\frac{x}{7}-9=0 \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Divide }\text{both }\text{sides }\text{by }15: \\
 x^2-\frac{x}{105}-\frac{3}{5}=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{3}{5} \text{to }\text{both }\text{sides}: \\
 x^2-\frac{x}{105}=\frac{3}{5} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{1}{44100} \text{to }\text{both }\text{sides}: \\
 x^2-\frac{x}{105}+\frac{1}{44100}=\frac{26461}{44100} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }\text{the }\text{left }\text{hand }\text{side }\text{as }\text{a }\text{square}: \\
 \left(x-\frac{1}{210}\right)^2=\frac{26461}{44100} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Take }\text{the }\text{square }\text{root }\text{of }\text{both }\text{sides}: \\
 x-\frac{1}{210}=\frac{\sqrt{26461}}{210}\text{ or }x-\frac{1}{210}=-\frac{\sqrt{26461}}{210} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{1}{210} \text{to }\text{both }\text{sides}: \\
 \fbox{$x=\frac{1}{210}+\frac{\sqrt{26461}}{210}$}\text{ or }x-\frac{1}{210}=-\frac{\sqrt{26461}}{210} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{1}{210} \text{to }\text{both }\text{sides}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & x=\frac{1}{210}+\frac{\sqrt{26461}}{210}\text{ or }\fbox{$x=\frac{1}{210}-\frac{\sqrt{26461}}{210}$} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/algebra/quadratic_roots_w_steps/988.txt'}