Answer
$x=\left\{ 2,3 \right\}$
Work Step by Step
Using the properties of equality, the given equation, $
x=\sqrt{x-2}+2
,$ is equivalent to
\begin{array}{l}\require{cancel}
x-2=\sqrt{x-2}
.\end{array}
Raising both sides to the second power, then the solution/s to the equation above is/are
\begin{array}{l}\require{cancel}
(x-2)^2=x-2
\\\\
(x)^2+2(x)(-2)+(-2)^2=x-2
\\\\
x^2-4x+4=x-2
\\\\
x^2+(-4x-x)+(4+2)=0
\\\\
x^2-5x+6=0
\\\\
(x-2)(x-3)=0
\\\\
x=\left\{ 2,3 \right\}
.\end{array}
Upon checking, both solutions $
x=\left\{ 2,3 \right\}
$ satisfy the original equation.