How to solve higher grade polynomials of complex numbers $q^{10}-2q^5+2=0$

Question

If I wanted to find the roots for $q^{10}-2q^5+2=0$, how would I go about doing that?

I tried treating it like a quadratic equation, but couldn't get there. I also tried putting $q=(a+ib)$ but that didn't do much.

Thank you in advance

Answer 1

You have to solve it in two steps.

Let $x=q^{5}$.

Then $x²-2x+2=0$

Solve this, and then solve the first equation to find the solutions in $q$.

Answer 2

Hint:

Substitute $q^5$ with $r$, so you get the equation $r^2-2r+2=0$