Let $a, b, c \in R,$ $a \ne 0$, such that $a$ and $4a+3b+2c$ have the same sign. Show that the equation $ax^2+bx+c=0$ cannot have both roots in the interval $(1,2)$.
Source: 1996 Romania NMO IX p1
Tags: algebra, inequalities
Let $a, b, c \in R,$ $a \ne 0$, such that $a$ and $4a+3b+2c$ have the same sign. Show that the equation $ax^2+bx+c=0$ cannot have both roots in the interval $(1,2)$.