Answer
$x= 12 $
Work Step by Step
Given \begin{equation}
5 \sqrt{x-8}=\sqrt{x+4}+6.
\end{equation} First rearrange the equation and square both sides to eliminate the radical signs (may be repeated), and solve for $x$.
\begin{equation}
\begin{aligned}
5 \sqrt{x-8}&=\sqrt{x+4}+6\\
\left(5 \sqrt{x-8}\right)^2&= \left( \sqrt{x+4}+6 \right)^2\\
25x-200& = x+4+12\sqrt{x+4}+36 \\
24x-240&= 12\sqrt{x+4}\\
\left(24x-240 \right)^2& = \left( 12\sqrt{x+4} \right)^2 \\
576x^2-11520x+57600&= 144x+576\\
576x^2-11664x+57024&=0.\\
a & =576, b=-11664, c=57024 \\
x & =\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \\
x & =\frac{-(-11664) \pm \sqrt{(-11664)^2-4 \cdot 576 \cdot(57024)}}{2 \cdot 576} \\
& =\frac{11664 \pm 2160}{1152} \\
\Longrightarrow x& =\frac{11664 + 2160}{1152}\\
&= 12\\
x& =\frac{11664 - 2160}{1152}\\
&= \frac{33}{4}.
\end{aligned}
\end{equation} Check. \begin{equation}
\begin{aligned}
5 \sqrt{33/4-8}&\stackrel{?}{=}\sqrt{33/4+4}+6\\
2.5& =9.5\quad \quad \textbf{False}\\
5 \sqrt{12-8}&\stackrel{?}{=}\sqrt{12+4}+6\\
10& =10\quad \quad \textbf{True}
\end{aligned}
\end{equation} The solution is $x= 12$.
