Problem:
Solve $\sqrt{5 x+12}+\sqrt{6 x+4}=3$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Solve for }x: \\
 \sqrt{5 x+12}+\sqrt{6 x+4}=3 \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \left(\sqrt{5 x+12}+\sqrt{6 x+4}\right)^2=16+11 x+2 \sqrt{5 x+12} \sqrt{6 x+4}=16+11 x+2 \sqrt{(5 x+12) (6 x+4)}=9: \\
 16+11 x+2 \sqrt{(5 x+12) (6 x+4)}=9 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }11 x+16 \text{from }\text{both }\text{sides}: \\
 2 \sqrt{(5 x+12) (6 x+4)}=-11 x-7 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Raise }\text{both }\text{sides }\text{to }\text{the }\text{power }\text{of }\text{two}: \\
 4 (5 x+12) (6 x+4)=(-11 x-7)^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+368 x+192=(-11 x-7)^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+368 x+192=121 x^2+154 x+49 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }121 x^2+154 x+49 \text{from }\text{both }\text{sides}: \\
 -x^2+214 x+143=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply }\text{both }\text{sides }\text{by }-1: \\
 x^2-214 x-143=0 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\text{143 }\text{to }\text{both }\text{sides}: \\
 x^2-214 x=143 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\text{11449 }\text{to }\text{both }\text{sides}: \\
 x^2-214 x+11449=11592 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }\text{the }\text{left }\text{hand }\text{side }\text{as }\text{a }\text{square}: \\
 (x-107)^2=11592 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Take }\text{the }\text{square }\text{root }\text{of }\text{both }\text{sides}: \\
 x-107=6 \sqrt{322}\text{ or }x-107=-6 \sqrt{322} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\text{107 }\text{to }\text{both }\text{sides}: \\
 \fbox{$x=107+6 \sqrt{322}$}\text{ or }x-107=-6 \sqrt{322} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\text{107 }\text{to }\text{both }\text{sides}: \\
 x=107+6 \sqrt{322}\text{ or }\fbox{$x=107-6 \sqrt{322}$} \\
\end{array}
 \\
 
\begin{array}{l}
 \sqrt{5 x+12}+\sqrt{6 x+4} \text{$\Rightarrow $ }\sqrt{5 \left(107-6 \sqrt{322}\right)+12}+\sqrt{6 \left(107-6 \sqrt{322}\right)+4} \text{= }\sqrt{5 \left(107-6 \sqrt{322}\right)+12}+\sqrt{6 \left(107-6 \sqrt{322}\right)+4} \text{$\approx $ }3.: \\
 \text{So }\text{this }\text{solution }\text{is }\text{correct} \\
\end{array}
 \\
 
\begin{array}{l}
 \sqrt{5 x+12}+\sqrt{6 x+4} \text{$\Rightarrow $ }\sqrt{5 \left(6 \sqrt{322}+107\right)+12}+\sqrt{6 \left(6 \sqrt{322}+107\right)+4} \text{= }\sqrt{5 \left(6 \sqrt{322}+107\right)+12}+\sqrt{6 \left(6 \sqrt{322}+107\right)+4} \text{$\approx $ }68.8887: \\
 \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=107-6 \sqrt{322} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/algebra/sqrt_equations_w_steps/4506.txt'}