Question:
Solve the equation $\sqrt{2} x^{2}+x+\sqrt{2}=0$
Solution:
The given quadratic equation is $\sqrt{2} x^{2}+x+\sqrt{2}=0$
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, the discriminant of the given equation is
$D=b^{2}-4 a c=1^{2}-4 \times \sqrt{2} \times \sqrt{2}=1-8=-7$
Therefore, the required solutions are
$\frac{-b \pm \sqrt{\mathrm{D}}}{2 a}=\frac{-1 \pm \sqrt{-7}}{2 \times \sqrt{2}}=\frac{-1 \pm \sqrt{7} i}{2 \sqrt{2}}$ $[\sqrt{-1}=i]$