Answer
The solution for the equation $\sqrt{x+7}+5=x$ is $\left\{ 9 \right\}$.
Work Step by Step
Consider the equation $\sqrt{x+7}+5=x$.
Simplify the the provided equation
$\begin{align}
& \sqrt{x+7}+5=x \\
& \sqrt{x+7}=\left( x-5 \right)
\end{align}$.
Square both sides of the equation.
$\begin{align}
& {{\left( \sqrt{x+7} \right)}^{2}}={{\left( x-5 \right)}^{2}} \\
& x+7={{x}^{2}}-10x+25 \\
\end{align}$
Collect all variables and constants on one side.
$\begin{align}
& x+7={{x}^{2}}-10x+25 \\
& 0={{x}^{2}}-10x+25-x-7 \\
& 0={{x}^{2}}-11x+18
\end{align}$
So, the equation is:
${{x}^{2}}-11x+18=0$
Factorize the equation to obtain the solution.
${{x}^{2}}-11x+18=0$
Choose the factors of $18$ , so that the sum of the factors is $-11$.
The required factors are $2\text{ and }9$.
$\left( x-9 \right)\left( x-2 \right)=0$
This gives;
$x=2\text{ and }x=9$
Check the obtained solution by substitution of $x=9$ in the equation $\sqrt{x+7}+5=x$.
$\begin{align}
& \sqrt{x+7}+5=x \\
& \sqrt{9+7}+5\overset{?}{\mathop{=}}\,9 \\
& \text{ }4+5\overset{?}{\mathop{=}}\,9 \\
& \text{ }9\overset{?}{\mathop{=}}\,9 \\
\end{align}$
This is true.
Substitute $x=2$ in the equation $\sqrt{x+7}+5=x$.
$\begin{align}
& \sqrt{x+7}+5=x \\
& \sqrt{2+7}+5\overset{?}{\mathop{=}}\,2 \\
& \text{ }3+5\overset{?}{\mathop{=}}\,2 \\
& \text{ }8\ne 2 \\
\end{align}$
This is false.
Hence, the value $x=9$ satisfies the above equation.
Thus, the solution set is $\left\{ 9 \right\}$.
