Solve the following

$-x^{2}+x-2=0$

$\Rightarrow x^{2}-x+2=0$

$\Rightarrow x^{2}-x+\frac{1}{4}+\frac{7}{4}=0$

$\Rightarrow x^{2}-2 \times x \times \frac{1}{2}+\left(\frac{1}{2}\right)^{2}-\frac{7}{4} i^{2}=0$

$\Rightarrow\left(x-\frac{1}{2}\right)^{2}-\left(\frac{i \sqrt{7}}{2}\right)^{2}=0$

$\Rightarrow\left(x-\frac{1}{2}+\frac{i \sqrt{7}}{2}\right)\left(x-\frac{1}{2}-\frac{i \sqrt{7}}{2}\right)=0$

$\Rightarrow\left(x-\frac{1}{2}+\frac{\sqrt{7}}{2} i\right)=0$ or, $\left(x-\frac{1}{2}-\frac{\sqrt{7}}{2} i\right)=0$

$\Rightarrow x=\frac{1}{2}-\frac{\sqrt{7}}{2} i$ or, $x=\frac{1}{2}+\frac{\sqrt{7}}{2} i$

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