#### Answer

$\color{blue}{\left\{\dfrac{3-i\sqrt3}{5}, \dfrac{3+i\sqrt3}{5}\right\}}$.

#### Work Step by Step

RECALL:
If $x^2=a$, then taking the square root of both sides gives $x = \pm \sqrt{a}$.
Take the square root of both sides of the given equation to obtain:
$\sqrt{(5x-3)^2}=\pm \sqrt{-3}
\\5x-3 =\pm \sqrt{-1(3)}$
Since $\sqrt{-1}=i$, then the expression above is equivalent to:
$5x-3=\pm i\sqrt{3}$
Add $3$ to both sides:
$x =3 \pm i\sqrt{3}$
Divide $5$ to both sides:
$x=\dfrac{3\pm i\sqrt3}{5}$
Thus, the solution set is $\color{blue}{\left\{\dfrac{3-i\sqrt3}{5}, \dfrac{3+i\sqrt3}{5}\right\}}$.