#### Answer

$x=\left\{ 1-2\sqrt{2},1+2\sqrt{2} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
(x-1)^2=8
,$ take the square root of both sides and simplify the resulting radical. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Taking the square root of both sides (Square Root Property), the equation above is equivalent to
\begin{array}{l}\require{cancel}
x-1=\pm\sqrt{8}
.\end{array}
Extracting the root of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
x-1=\pm\sqrt{4\cdot2}
\\\\
x-1=\pm\sqrt{(2)^2\cdot2}
\\\\
x-1=\pm2\sqrt{2}
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
x=1\pm2\sqrt{2}
.\end{array}
Hence the solutions are $
x=\left\{ 1-2\sqrt{2},1+2\sqrt{2} \right\}
.$