Problem

Source:

Tags: AMC, USA(J)MO, USAMO, algebra, polynomial, Vieta, algebra unsolved



The cubic equation $x^3 + ax^2 + bx + c = 0$ has three real roots. Show that $a^2-3b\geq 0$, and that $\sqrt{a^2-3b}$ is less than or equal to the difference between the largest and smallest roots.