Question:
$\sqrt{5} x^{2}+x+\sqrt{5}=0$
Solution:
Given:
$\sqrt{5} \mathrm{x}^{2}+\mathrm{x}+\sqrt{5}=0$
Solution of a general quadratic equation $a x^{2}+b x+c=0$ is given by:
$x=\frac{-b \pm \sqrt{\left(b^{2}-4 a c\right)}}{2 a}$
$\Rightarrow x=\frac{-1 \pm \sqrt{(1)^{2}-(4 \times \sqrt{5} \times \sqrt{5})}}{2 \times \sqrt{5}}$
$\Rightarrow x=\frac{-1 \pm \sqrt{1-20}}{2 \sqrt{5}}$
$x=\frac{-1 \pm \sqrt{-19}}{2 \sqrt{5}}$
$\Rightarrow \quad x=\frac{-1 \pm \sqrt{19} i}{2 \sqrt{5}}$
$\Rightarrow x=-\frac{1}{2 \sqrt{5}} \pm \frac{\sqrt{19}}{2 \sqrt{5}} i$
Ans: $x=-\frac{\sqrt{5}}{10}+\frac{\sqrt{\frac{19}{5}}}{2} i$ and $x=-\frac{\sqrt{5}}{10}-\frac{\sqrt{\frac{19}{5}}}{2} i$