If z be any complex number and if $z^2 + az + b = 0$ has two roots both of which has unit modulus, then prove that $|a| \leq 2$, and $|b| = 1$

Question

If $z$ be any complex number and if $z^2 + az + b = 0$ has two roots both of which has unit modulus, then prove that $|a| \leq 2, |b| = 1$

I'm using the quadratic equation formula to find the roots and then equating it's modulus to $1$ but I don't understand how to approach after that to prove the given result.

Answer 1

Viète's formulas are also valid for complex numbers. Since both roots are of unit modulus, their product $b$ also has unit modulus, while their sum $-a$ has modulus at most 2 by the triangle inequality, so $|a|\le2$.