#### Answer

$x=\left\{ 5-6i,5+6i \right\}$

#### Work Step by Step

Taking the square root of both sides (the Square Root Property) and using $i=\sqrt{-1}$, the solutions to the given equation, $
(x-5)^2=-36
,$ are
\begin{array}{l}\require{cancel}
x-5=\pm\sqrt{-36}
\\\\
x-5=\pm\sqrt{-1}\cdot\sqrt{36}
\\\\
x-5=\pm i\cdot\sqrt{(6)^2}
\\\\
x-5=\pm i\cdot6
\\\\
x-5=\pm 6i
\\\\
x=5\pm 6i
.\end{array}
Hence, $
x=\left\{ 5-6i,5+6i \right\}
.$