Problem:
Solve $\sqrt{2 x+5}+\sqrt{15 x-6}=14$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Solve for }x: \\
 \sqrt{2 x+5}+\sqrt{15 x-6}=14 \\
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 \\
\hline
 
\begin{array}{l}
 \left(\sqrt{2 x+5}+\sqrt{15 x-6}\right)^2=-1+17 x+2 \sqrt{2 x+5} \sqrt{15 x-6}=-1+17 x+2 \sqrt{(2 x+5) (15 x-6)}=196: \\
 -1+17 x+2 \sqrt{(2 x+5) (15 x-6)}=196 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }17 x-1 \text{from }\text{both }\text{sides}: \\
 2 \sqrt{(2 x+5) (15 x-6)}=197-17 x \\
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 \\
 
\begin{array}{l}
 \text{Raise }\text{both }\text{sides }\text{to }\text{the }\text{power }\text{of }\text{two}: \\
 4 (2 x+5) (15 x-6)=(197-17 x)^2 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Expand }\text{out }\text{terms }\text{of }\text{the }\text{left }\text{hand }\text{side}: \\
 120 x^2+252 x-120=(197-17 x)^2 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Expand }\text{out }\text{terms }\text{of }\text{the }\text{right }\text{hand }\text{side}: \\
 120 x^2+252 x-120=289 x^2-6698 x+38809 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }289 x^2-6698 x+38809 \text{from }\text{both }\text{sides}: \\
 -169 x^2+6950 x-38929=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide }\text{both }\text{sides }\text{by }-169: \\
 x^2-\frac{6950 x}{169}+\frac{38929}{169}=0 \\
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 \\
 
\begin{array}{l}
 \text{Subtract }\frac{38929}{169} \text{from }\text{both }\text{sides}: \\
 x^2-\frac{6950 x}{169}=-\frac{38929}{169} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{12075625}{28561} \text{to }\text{both }\text{sides}: \\
 x^2-\frac{6950 x}{169}+\frac{12075625}{28561}=\frac{5496624}{28561} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }\text{the }\text{left }\text{hand }\text{side }\text{as }\text{a }\text{square}: \\
 \left(x-\frac{3475}{169}\right)^2=\frac{5496624}{28561} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Take }\text{the }\text{square }\text{root }\text{of }\text{both }\text{sides}: \\
 x-\frac{3475}{169}=\frac{84 \sqrt{779}}{169}\text{ or }x-\frac{3475}{169}=-\frac{84 \sqrt{779}}{169} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{3475}{169} \text{to }\text{both }\text{sides}: \\
 \fbox{$x=\frac{3475}{169}+\frac{84 \sqrt{779}}{169}$}\text{ or }x-\frac{3475}{169}=-\frac{84 \sqrt{779}}{169} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{3475}{169} \text{to }\text{both }\text{sides}: \\
 x=\frac{3475}{169}+\frac{84 \sqrt{779}}{169}\text{ or }\fbox{$x=\frac{3475}{169}-\frac{84 \sqrt{779}}{169}$} \\
\end{array}
 \\
 
\begin{array}{l}
 \sqrt{2 x+5}+\sqrt{15 x-6} \text{$\approx $ }14.: \\
 \text{So }\text{this }\text{solution }\text{is }\text{correct} \\
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 \\
 
\begin{array}{l}
 \sqrt{2 x+5}+\sqrt{15 x-6} \text{$\approx $ }31.1895: \\
 \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{3475}{169}-\frac{84 \sqrt{779}}{169} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/algebra/sqrt_equations_w_steps/726.txt'}