[Math] Find complex roots of $x^2+x+1$

Question

I'm quite inept when it comes to calculations with complex numbers, and my task is to find complex roots of $x^2+x+1$. Well obviously they are $x_1=-\dfrac{1}{2}-\dfrac{i\sqrt{3}}{2}$ and $x_2=-\dfrac{1}{2}+\dfrac{i\sqrt{3}}{2}$. But how come when I put the quadratic function into WolphramAlpha it gives back $-\sqrt[3]{-1}$ (cubic root) and $(-1)^{2/3}$. How could I deduce it on my own?

Answer 1

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

Kwadraat afsplitsen in Dutch. I don't know the English equivalent for it.