A question regarding the solution of a quadratic polynomial

Question

I'd like to know why the following statement (taken from Iteration of Rational Functions by Beardon) is true: "Now $z^2+c$ has two fixed points, say $\alpha$ and $\beta$, in $\mathbb{C}$, and as these are solutions of $$z^2-z+c=0,$$ they satisfy $$\alpha+\beta=1,\quad\alpha\beta=c."$$ I know this seems trivial but the answer is escaping me.

Answer 1

These are Vieta's formulas. If $\alpha$ and $\beta$ are roots, then by the Factor Theorem, $(z-\alpha)$ and $(z-\beta)$ are factors, so $z^2-z+c=(z-\alpha)(z-\beta)$. Equating coefficients yields $\alpha+\beta=1$ and $\alpha\beta=c$.