Answer
$x=-\dfrac{9}{2}\pm\dfrac{\sqrt{105}}{2}$
Work Step by Step
$\dfrac{x^{2}}{2}-3=-\dfrac{9}{2}x$
Multiply the whole equation by $2$ to avoid working with fractions:
$2\Big(\dfrac{x^{2}}{2}-3=-\dfrac{9}{2}x\Big)$
$x^{2}-6=-9x$
Take the $-9x$ to the left side of the equation:
$x^{2}+9x-6=0$
Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a=1$, $b=9$ and $c=-6$
Substitute:
$x=\dfrac{-9\pm\sqrt{9^{2}-4(1)(-6)}}{2(1)}=\dfrac{-9\pm\sqrt{81+24}}{2}=...$
$...=\dfrac{-9\pm\sqrt{105}}{2}=-\dfrac{9}{2}\pm\dfrac{\sqrt{105}}{2}$
