Does closeness of coefficients of two quadratic equations imply the closeness of roots

Question

Let $x^2+ax+b=0$ and $x^2+a'x+b'=0$ be two quadratic polynomials with complex coefficients. Let $(z_1,z_2)$ and$(z_1',z_2')$ be the roots of the first and second equations respectively. If $\epsilon>0$ is given, does there exist a $\delta>0$, such that $|a-a'|+|b-b'|<\delta$ implies that $|z_1-z_1'|+|z_2-z_2'|<\epsilon$. Can we somehow apply the fundamental theorem oof algebra to prove the same?

Answer 1

Yes, this is true and a well-known fact. Expressed in a suitable way, it is true that the roots of a polynomial of fixed degree are a continuous function of the coefficients. And so for monic quadratics, this means pretty much exactly what you wrote (up to re-labelling $z_1' \leftrightarrow z_2'$).

See this question: The roots of a polynomial are a continuous function of the coefficients