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

This equation can also be written as $2 x^{2}-4 x+3=0$

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

we obtain $a=2, b=-4$ and $c=3$

Therefore, the discriminant of the given equation is

$D=b^{2}-4 a c=(-4)^{2}-4 \times 2 \times 3=16-24=-8$

Therefore, the required solutions are

$\frac{-b \pm \sqrt{D}}{2 a}=\frac{-(-4) \pm \sqrt{8}}{2 \times 2}=\frac{4 \pm 2 \sqrt{2} i}{4} \quad[\sqrt{-1}=i]$

$=\frac{2 \pm \sqrt{2} i}{2}=1 \pm \frac{\sqrt{2}}{2} i$