Question

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.

Answer 1

assume a, b, c >0, then prove this inequality

this is enough to guarantee that there will be one root between roots of those first two integrals.