For f(x) = $ax^2$ + bx + c:

Case 1: D < 0 (no real roots)

• a > 0: f(x) > 0 for all x (always positive)
• a < 0: f(x) < 0 for all x (always negative)

Case 2: D = 0 (one repeated root r)

• a > 0: f(x) >= 0, equals 0 only at x = r
• a < 0: f(x) <= 0, equals 0 only at x = r

Case 3: D > 0 (two real roots alpha < beta)

• a > 0: f(x) < 0 for alpha < x < beta, f(x) > 0 outside
• a < 0: f(x) > 0 for alpha < x < beta, f(x) < 0 outside

Key application: $ax^2$ + bx + c > 0 for all x iff a > 0 and D < 0. This condition appears frequently in JEE.