Answer
The solution set of the equation $\sqrt{x+8}-\sqrt{x-4}=2$ is $\left\{ 8 \right\}$.
Work Step by Step
The provided equation is $\sqrt{x+8}-\sqrt{x-4}=2$.
Add $\sqrt{x-4}$ on both sides.
$\begin{align}
& \sqrt{x+8}-\sqrt{x-4}+\sqrt{x-4}=\sqrt{x-4}+2 \\
& \sqrt{x+8}=\sqrt{x-4}+2
\end{align}$
Take the square on both sides.
$\begin{align}
& {{\left( \sqrt{x+8} \right)}^{2}}={{\left( \sqrt{x-4}+2 \right)}^{2}} \\
& \left( x+8 \right)={{\left( \sqrt{x-4} \right)}^{2}}+2\left( \sqrt{x-4} \right)\left( 2 \right)+{{2}^{2}} \\
& x+8=x-4+4\sqrt{x-4}+4 \\
& x+8=x+4\sqrt{x-4}
\end{align}$
Subtract $x$ from both sides.
$\begin{align}
& x+8-x=x+4\sqrt{x-4}-x \\
& 8=4\sqrt{x-4}
\end{align}$
Divide $4$ on both sides.
$2=\sqrt{x-4}$
Take the square on both sides.
$4=x-4$
Add $4$ on both sides.
$\begin{align}
& 4+4=x-4+4 \\
& x=8
\end{align}$
The solution set of the provided equation $\sqrt{x+8}-\sqrt{x-4}=2$ is $\left\{ 8 \right\}$.
