Folklore

The coefficients of the quadratic equation $x^{2}+p x+q=0$ have changed by no more than 0.001.

Can the larger root of the equation change by more than 1000?
Make the coefficient $p$ very large.

## Solution

Consider the equations $x^{2}-2 p x+p^{2}=0$ and $x^{2}-2(p+a) x+p^{2}=0$, where $p=10^{9}, a=5 \cdot 10^{-4}$. The first equation has a single root $p$, and the larger root of the second equation is

$(p+a)+\sqrt{p^{2}+2 a p+a^{2}-p^{2}}>p+\sqrt{2 a p}=p+1000$.

## Answer

Yes.