Solve the following

We have:

$x^{2}-x+1=0$

$\Rightarrow x^{2}-x+\frac{1}{2}+\frac{3}{4}=0$

$\Rightarrow x^{2}-2 \times x \times \frac{1}{2}+\left(\frac{1}{2}\right)^{2}-\frac{3}{4} i^{2}=0$

$\Rightarrow\left(x-\frac{1}{2}\right)^{2}-\left(\frac{i \sqrt{3}}{2}\right)^{2}=0$

$\Rightarrow\left(x-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)\left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)=0$

$\Rightarrow\left(x-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)=0 \quad$ or $\quad\left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)=0$

$\Rightarrow x=\frac{1}{2}-\frac{i \sqrt{3}}{2} \quad$ or $\quad x=\frac{1}{2}+\frac{i \sqrt{3}}{2}$

Hence, the roots of the equation are $\frac{1}{2} \pm \frac{i \sqrt{3}}{2}$.