#### Answer

$x=\left\{ \dfrac{1-2i\sqrt{2}}{6},\dfrac{1+2i\sqrt{2}}{6} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To find the non real complex solutions of the given equation, $
(6x-1)^2=-8
,$ take the square root of both sides (Square Root Property). Then use the properties of radicals and use $i=\sqrt{-1}.$ Finally, simplify the radical and 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}
6x-1=\pm\sqrt{-8}
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the equation above is equivalent to\begin{array}{l}\require{cancel}
6x-1=\pm\sqrt{-1}\cdot\sqrt{8}
.\end{array}
Using $i=\sqrt{-1},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
6x-1=\pm i\sqrt{8}
.\end{array}
Writing the radicand as an expression that contains a factor that is a perfect power of the given index and then extracting the root of that factor, the equation above is equivalent to
\begin{array}{l}\require{cancel}
6x-1=\pm i\sqrt{4\cdot2}
\\\\
6x-1=\pm i\sqrt{(2)^2\cdot2}
\\\\
6x-1=\pm i(2)\sqrt{2}
\\\\
6x-1=\pm 2i\sqrt{2}
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
6x=1\pm 2i\sqrt{2}
\\\\
x=\dfrac{1\pm 2i\sqrt{2}}{6}
.\end{array}
Hence, $
x=\left\{ \dfrac{1-2i\sqrt{2}}{6},\dfrac{1+2i\sqrt{2}}{6} \right\}
.$