ISL 1989 Problem 77

Let $a, b, c, r,$ and $s$ be real numbers. Show that if $r$ is a root of $ax^2+bx+c = 0$ and s is a root of $-ax^2+bx+c = 0,$ then $$\frac{a}{2} x^2 + bx + c = 0$$ has a root between $r$ and $s.$