Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$
Source: 2008 Cuba 2.1
Tags: algebra, inequalities, trinomial
Given a polynomial of degree $2$, $p(x) = ax^2 +bx+c$ define the function $$S(p) = (a -b)^2 + (b - c)^2 + (c - a)^2.$$Determine the real number$ r$such that, for any polynomial $p(x)$ of degree $2$ with real roots, holds $S(p) \ge ra^2$