Let
and
be two positive real numbers. If
is a real solution of the equation
with real coefficients
and
such that
prove that
Conversely, if
satisfies the above inequality, prove that there exist real numbers
and
with
such that
is one of the roots of the equation
%V0
$(CZS 3)$ 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 $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, 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 $x^2+px+ q = 0.$
Školjka