Question 14
(CZS 3) Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2+px+q=0$ with real coefficients $p$ and $q$ such that $|p|\le a$, $|q|\le b$, prove that
$$|x|\le\frac12\left(a+\sqrt{a^2+4b}\right).\tag{1}$$
Conversely, if $x$ satisfies (1), prove that there exist real numbers $p$ and $q$ with $|p|\le a$, $|q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+q=0$.
$$|x|\le\frac12\left(a+\sqrt{a^2+4b}\right).\tag{1}$$
Conversely, if $x$ satisfies (1), prove that there exist real numbers $p$ and $q$ with $|p|\le a$, $|q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+q=0$.