Answer
$\{-2,4\}$
Work Step by Step
We are given the equation:
$$\begin{align}(x-1)^2-9=0.\end{align}$$
Graph $y=(x-1)^2-9$. From the graph we find the $x$-intercepts:
$$\begin{align*}
x_1&=-2\\
x_2&=4.
\end{align*}$$
We check the solutions by substituting them in Eq. $(1)$:
$$\begin{align*}
x_1&=-2\\
(-2-1)^2-9&\stackrel{?}{=}0\\
9-9&\stackrel{?}{=}0\\
0&=0\checkmark\\\\
x_2&=4\\
(4-1)^2-9&\stackrel{?}{=}0\\
9-9&\stackrel{?}{=}0\\
0&=0\checkmark\\
\end{align*}$$
Both solutions check the equation, so the solution set is:
$$\{-2,4\}.$$
