$ax^{2+bx+c}$ > 0 for ALL x iff a > 0 AND D < 0. $ax^{2+bx+c}$ < 0 for ALL x iff a < 0 AND D < 0. These conditions are crucial in domain/range problems: sqrt($ax^{2+bx+c}$) is defined for all x only when the quadratic is non-negative everywhere. Similarly, log($ax^{2+bx+c}$) requires the quadratic to be always positive.