An Estimate for the Roots of Quadratic Equation

Solution

Observe that

\displaystyle\begin{align}\left|\Delta\right|&=\left|b^2-4ac\right|\le\left|b^2\right|+\left|-4ac\right|\\ &\le\left|b\right|^2+4\left|ac\right|\le\left(\left|b\right|+2\frac{|ac|}{|b|}\right)^2, \end{align}

with equality only when $4|ac|^2=0,$ which, together with the given condition $a\ne 0,$ means that $|c|=0,$ i.e., $c=0.$ It follows, from the quadratic formula, that

\displaystyle \begin{align} |z|&=\left|\frac{-b\pm\sqrt{\Delta}}{2a}\right|\le\frac{|-b|+\sqrt{\Delta}}{2|a|}\\ &\le\frac{\displaystyle |b|+\left(|b|+2\frac{|ac|}{|b|}\right)}{2|a|}=\frac{\displaystyle |b|+\frac{|a|\cdot |c|}{|b|}}{|a|}\\ &=\left|\frac{b}{a}\right|+\left|\frac{c}{b}\right|. \end{align}

The above means that the roots of the quadratic polynomial $az^2+bz+c$ lie in the disk, centered at the origin with the radius of $\displaystyle \left|\frac{b}{a}\right|+\left|\frac{c}{b}\right|.$

Acknowledgment

The problem was kindly posted at the CutTheKnotMath facebook page by Dorin Marghidanu, along with a solution of his.