Question:
Solve the equation $x^{2}+\frac{x}{\sqrt{2}}+1=0$
Solution:
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 this 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
$\frac{-b \pm \sqrt{D}}{2 a}=\frac{-1 \pm \sqrt{-7}}{2 \sqrt{2}}=\frac{-1 \pm \sqrt{7} i}{2 \sqrt{2}}$ $[\sqrt{-1}=i]$