Answer
$\left\{ \dfrac{-9-\sqrt{105}}{2},\dfrac{-9+\sqrt{105}}{2} \right\}$
Work Step by Step
The standard form of the given equation, $
\dfrac{x^2}{2}-3=-\dfrac{9}{2}x
,$ is
\begin{array}{l}\require{cancel}
2\left( \dfrac{x^2}{2}-3 \right)=\left( -\dfrac{9}{2}x \right)2
\\\\
1(x^2)-2(3)=-9x(1)
\\\\
x^2-6=-9x
\\\\
x^2+9x-6=0
.\end{array}
Using $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (or the Quadratic Formula), the solutions of the quadratic equation, $
x^2+9x-6=0
,$ are
\begin{array}{l}\require{cancel}
\dfrac{-(9)\pm\sqrt{(9)^2-4(1)(-6)}}{2(1)}
\\\\=
\dfrac{-9\pm\sqrt{81+24}}{2}
\\\\=
\dfrac{-9\pm\sqrt{105}}{2}
.\end{array}
Hence, the solutions are $
\left\{ \dfrac{-9-\sqrt{105}}{2},\dfrac{-9+\sqrt{105}}{2} \right\}
.$
