Solve the following

Given equation: $x^{2}+\frac{x}{\sqrt{2}}+1=0$

Comparing the given equation with the general form of the quadratic equation $a x^{2}+b x+c=0$, we get $a=1, b=\frac{1}{\sqrt{2}}$ and $c=1$.

Substituting these values in $\alpha=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}$ and $\beta=\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}$, we get:

$\alpha=\frac{-\frac{1}{\sqrt{2}}+\sqrt{\frac{1}{2}-4 \times 1 \times 1}}{2} \quad$ and $\quad \beta=\frac{-\frac{1}{\sqrt{2}}-\sqrt{\frac{1}{2}-4 \times 1 \times 1}}{2}$

$\alpha=\frac{-\frac{1}{\sqrt{2}}+\sqrt{-\frac{7}{2}}}{2} \quad$ and $\quad \beta=\frac{-\frac{1}{\sqrt{2}}-\sqrt{-\frac{7}{2}}}{2}$

$\alpha=\frac{-\frac{1}{\sqrt{2}}+i \sqrt{\frac{7}{2}}}{2} \quad$ and $\quad \beta=\frac{-\frac{1}{\sqrt{2}}-i \sqrt{\frac{7}{2}}}{2}$

$\alpha=\frac{-1+i \sqrt{7}}{2 \sqrt{2}} \quad$ and $\quad \beta=\frac{-1-i \sqrt{7}}{2 \sqrt{2}}$

Hence, the roots of the equation are $\frac{-1 \pm i \sqrt{7}}{2 \sqrt{2}}$.