Problem:
Solve $\sqrt{6 x+8}+\sqrt{15 x+10}=10$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Solve for }x: \\
 \sqrt{6 x+8}+\sqrt{15 x+10}=10 \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \left(\sqrt{6 x+8}+\sqrt{15 x+10}\right)^2=18+21 x+2 \sqrt{6 x+8} \sqrt{15 x+10}=18+21 x+2 \sqrt{(6 x+8) (15 x+10)}=100: \\
 18+21 x+2 \sqrt{(6 x+8) (15 x+10)}=100 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }21 x+18 \text{from }\text{both }\text{sides}: \\
 2 \sqrt{(6 x+8) (15 x+10)}=82-21 x \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Raise }\text{both }\text{sides }\text{to }\text{the }\text{power }\text{of }\text{two}: \\
 4 (6 x+8) (15 x+10)=(82-21 x)^2 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Expand }\text{out }\text{terms }\text{of }\text{the }\text{left }\text{hand }\text{side}: \\
 360 x^2+720 x+320=(82-21 x)^2 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Expand }\text{out }\text{terms }\text{of }\text{the }\text{right }\text{hand }\text{side}: \\
 360 x^2+720 x+320=441 x^2-3444 x+6724 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }441 x^2-3444 x+6724 \text{from }\text{both }\text{sides}: \\
 -81 x^2+4164 x-6404=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide }\text{both }\text{sides }\text{by }-81: \\
 x^2-\frac{1388 x}{27}+\frac{6404}{81}=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }\frac{6404}{81} \text{from }\text{both }\text{sides}: \\
 x^2-\frac{1388 x}{27}=-\frac{6404}{81} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{481636}{729} \text{to }\text{both }\text{sides}: \\
 x^2-\frac{1388 x}{27}+\frac{481636}{729}=\frac{424000}{729} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }\text{the }\text{left }\text{hand }\text{side }\text{as }\text{a }\text{square}: \\
 \left(x-\frac{694}{27}\right)^2=\frac{424000}{729} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Take }\text{the }\text{square }\text{root }\text{of }\text{both }\text{sides}: \\
 x-\frac{694}{27}=\frac{40 \sqrt{265}}{27}\text{ or }x-\frac{694}{27}=-\frac{40 \sqrt{265}}{27} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{694}{27} \text{to }\text{both }\text{sides}: \\
 \fbox{$x=\frac{694}{27}+\frac{40 \sqrt{265}}{27}$}\text{ or }x-\frac{694}{27}=-\frac{40 \sqrt{265}}{27} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{694}{27} \text{to }\text{both }\text{sides}: \\
 x=\frac{694}{27}+\frac{40 \sqrt{265}}{27}\text{ or }\fbox{$x=\frac{694}{27}-\frac{40 \sqrt{265}}{27}$} \\
\end{array}
 \\
 
\begin{array}{l}
 \sqrt{6 x+8}+\sqrt{15 x+10} \text{$\approx $ }10.: \\
 \text{So }\text{this }\text{solution }\text{is }\text{correct} \\
\end{array}
 \\
 
\begin{array}{l}
 \sqrt{6 x+8}+\sqrt{15 x+10} \text{$\approx $ }45.0384: \\
 \text{So }\text{this }\text{solution }\text{is }\text{incorrect} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{solution }\text{is}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & x=\frac{694}{27}-\frac{40 \sqrt{265}}{27} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/algebra/sqrt_equations_w_steps/3228.txt'}