Answer
$x=6$
Work Step by Step
Using the properties of equality, the given equation, $
x-\sqrt[]{x-2}=4
,$ is equivalent to
\begin{array}{l}\require{cancel}
x-4=\sqrt[]{x-2}
.\end{array}
Raising both sides of the equation above to the second power, then the solution/s is/are
\begin{array}{l}\require{cancel}
(x-4)^2=x-2
\\\\
(x)^2+2(x)(-4)+(-4)^2=x-2
\\\\
x^2-8x+16=x-2
\\\\
x^2+(-8x-x)+(16+2)=0
\\\\
x^2-9x+18=0
\\\\
(x-3)(x-6)=0
\\\\
x=\{ 3,6 \}
.\end{array}
Upon checking, only $
x=6
$ satisfies the original equation.