Answer
$\left\{\dfrac{2}{3}\pm\dfrac{5i}{3}\right\}$
Work Step by Step
Taking the square root of both sides (Square Root Property), the given equation, $
(3x-2)^2=-25
,$ is equivalent to
\begin{align*}
3x-2&=\pm\sqrt{-25}
.\end{align*}
Using concepts of simplifying radicals, the equation above is equivalent to
\begin{align*}
3x-2&=\pm\sqrt{25\cdot(-1)}
\\
3x-2&=\pm\sqrt{25}\cdot\sqrt{-1}
\\
3x-2&=\pm5\cdot\sqrt{-1}
\\
3x-2&=\pm5i
&(\text{use }i=\sqrt{-1})
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
3x&=2\pm5i
\\\\
\dfrac{\cancel3x}{\cancel3}&=\dfrac{2\pm5i}{3}
\\\\
x&=\dfrac{2\pm5i}{3}
\\\\
x&=\dfrac{2}{3}\pm\dfrac{5i}{3}
.\end{align*}
Hence, the solution set of $
(3x-2)^2=-25
$ is $
\left\{\dfrac{2}{3}\pm\dfrac{5i}{3}\right\}
$.
