The given quadratic equation is $\sqrt{3} x^{2}-\sqrt{2} x+3 \sqrt{3}=0$

On comparing the given equation with $a x^{2}+b x+c=0$

We obtain $a=\sqrt{3}, b=-\sqrt{2},$ and $c=3 \sqrt{3}$

Therefore, the discriminant of the given equation is

$D=b^{2}-4 a c=(-\sqrt{2})^{2}-4(\sqrt{3})(3 \sqrt{3})=2-36=-34$

Therefore, the required solutions are

$\frac{-b \pm \sqrt{D}}{2 a}=\frac{-(-\sqrt{2}) \pm \sqrt{-34}}{2 \times \sqrt{3}}=\frac{\sqrt{2} \pm \sqrt{34} i}{2 \sqrt{3}} \quad[\sqrt{-1}=i]$