Problem #PRU-77989

Problem

The equations $$\begin{array}{}\text{(1)}& a{x}^2+bx+c=0\end{array}$$ and $$\begin{array}{}\text{(2)}& -a{x}^2+bx+c\end{array}$$ are given. Prove that if $x_1$ and $x_2$ are, respectively, any roots of the equations (1) and (2), then there is a root $x_3$ of the equation $\frac{1}{2}a{x}^2+bx+c$ such that either $x_1\le x_3\le x_2$ or $x_1\ge x_3\ge x_2$.