IMO 1978 LL GDR25

Origin: GDR

Problem

Consider a polynomial P(x) = ax2 + bx + c with a > 0 that has two real roots x1, x2. Prove that the absolute values of both roots are less than or equal to 1 if and only if a + b + c \geq0, a −b + c \geq0, and a −c \geq0.