Answer
$x=\left\{ -\dfrac{1}{2},\dfrac{1}{3} \right\}$
Work Step by Step
Using $z=x^{-1}$, the given equation, $
3x^{-2}-3x^{-1}-18=0
,$ is equivalent to
\begin{array}{l}\require{cancel}
3z^2-3z-18=0
\\\\
\dfrac{3z^2-3z-18}{3}=\dfrac{0}{3}
\\\\
z^2-z-6=0
\\\\
(z-3)(z+2)=0
\\\\
z=\{-2,3\}
.\end{array}
Since $z=x^{-1}$, then if $z=-2$, then,
\begin{array}{l}\require{cancel}
-2=x^{-1}
\\\\
-2=\dfrac{1}{x}
\\\\
-\dfrac{1}{2}=x
.\end{array}
Since $z=x^{-1}$, then if $z=3$, then,
\begin{array}{l}\require{cancel}
3=x^{-1}
\\\\
3=\dfrac{1}{x}
\\\\
\dfrac{1}{3}=x
.\end{array}
Hence, $
x=\left\{ -\dfrac{1}{2},\dfrac{1}{3} \right\}
.$
