![(CZS 3)](/media/m/8/0/b/80baa0b2248e6c44f8b2ab68eed3a8cd.png)
Let
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
be two positive real numbers. If
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
is a real solution of the equation
![x^2 + px + q = 0](/media/m/9/2/7/9277ad42c828d845b9a014d08e1a2237.png)
with real coefficients
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
such that
![|p| \le a, |q| \le b,](/media/m/0/6/a/06a2ae4c9b9e117a157ac8165e6ba772.png)
prove that
![|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})](/media/m/9/1/5/915948103a9b472a2a2384ca8e034185.png)
Conversely, if
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
satisfies the above inequality, prove that there exist real numbers
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
with
![|p|\le a, |q|\le b](/media/m/f/a/5/fa5cb2fcd57529580d06eac37db05b35.png)
such that
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
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 $x^2+px+ q = 0.$