Answer
The solution set of the equation $\sqrt{x+5}-\sqrt{x-3}=2$ is $\left\{ 4 \right\}$.
Work Step by Step
The provided equation is $\sqrt{x+5}-\sqrt{x-3}=2$.
Add $\sqrt{x-3}$ on both sides.
$\begin{align}
& \sqrt{x+5}-\sqrt{x-3}+\sqrt{x-3}=\sqrt{x-3}+2 \\
& \sqrt{x+5}=\sqrt{x-3}+2
\end{align}$
Take the square on both sides.
$\begin{align}
& {{\left( \sqrt{x+5} \right)}^{2}}={{\left( \sqrt{x-3}+2 \right)}^{2}} \\
& \left( x+5 \right)={{\left( \sqrt{x-3} \right)}^{2}}+2\left( \sqrt{x-3} \right)\left( 2 \right)+{{2}^{2}} \\
& x+5=x-3+4\sqrt{x-3}+4 \\
& x+5=x+1+4\sqrt{x-3}
\end{align}$
Subtract $x$ from both sides.
$\begin{align}
& x+5-x=x+1+4\sqrt{x-3}-x \\
& 5=1+4\sqrt{x-3}
\end{align}$
Subtract $1$ from both sides.
$\begin{align}
& 5-1=1+4\sqrt{x-3}-1 \\
& 4=4\sqrt{x-3}
\end{align}$
Divide $4$ on both sides.
$1=\sqrt{x-3}$
Take the square on both sides.
$1=x-3$
Add $3$ on both sides.
$\begin{align}
& 1+3=x-3+3 \\
& x=4
\end{align}$
The solution set of the provided equation $\sqrt{x+5}-\sqrt{x-3}=2$ is $\left\{ 4 \right\}$.
