Question

Solve the equation $x^2+x+\frac{1}{\sqrt{2}}=0$

Answer 1

The given quadratic equation is $x^2+x+\frac{1}{\sqrt{2}}=0$
This equation can also be written as $\sqrt{2}{x}^2+\sqrt{2}x+1=0$
On comparing the given equation with ${ax}^2+bx+c=0$,
we obtain $a=\sqrt{2},b=\sqrt{2},$ and $c=1$
Therefore, the discriminant $D=b^2-4ac={\left(\sqrt{2}\right)}^2-4\left(\sqrt{2}\right)\times 1=2-4\sqrt{2}$
Therefore, the required solutions are
$\frac{-b\pm \sqrt{D}}{2a}=\frac{-\sqrt{2}\pm \sqrt{2-4\sqrt{2}}}{2\times \sqrt{2}}=\frac{-\sqrt{2}\pm \sqrt{2(1-2\sqrt{2})}}{2\sqrt{2}}$
$=\left(\frac{-\sqrt{2}\pm \sqrt{2}\left(\sqrt{2\sqrt{2}-1}i\right)}{2\sqrt{2}}\right)$
$=\frac{-1\pm 1\left(\sqrt{2\sqrt{2}-1}\right)i}{2}$