The given quadratic equation is $x^{2}+\frac{x}{\sqrt{2}}+1=0$

This equation can also be written as $\sqrt{2} x^{2}+x+\sqrt{2}=0$

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

We obtain $a=\sqrt{2}, b=1,$ and $c=\sqrt{2}$

$\therefore $ Discriminant $(D)=b^{2}-4 a c=1^{2}-4 \times \sqrt{2} \times \sqrt{2}=1-8=-7$

Therefore, the required solutions are

$\frac{-b \pm \sqrt{D}}{2 a}=\frac{-1 \pm \sqrt{-7}}{2 \sqrt{2}}=\frac{-1 \pm \sqrt{7} i}{2 \sqrt{2}}$      $[\sqrt{-1}=i]$