Answer
$x=5$
Work Step by Step
Using the properties of equality, the given equation, $
\sqrt{2x-1}+2=x
,$ is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{2x-1}=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}
2x-1=(x-2)^2
\\\\
2x-1=(x)^2+2(x)(-2)+(-2)^2
\\\\
2x-1=x^2-4x+4
\\\\
-x^2+(2x+4x)+(-1-4)=0
\\\\
-x^2+6x-5=0
\\\\
x^2-6x+5=0
\\\\
(x-5)(x-1)=0
\\\\
x=\left\{ 1,5 \right\}
.\end{array}
Upon checking, only $
x=5
$ satisfies the original equation.