Question

If a, b, c ∈ R, a ≠ 0 and the quadratic equation ax² + bx + c = 0 has no real root, then show that (a + b + c) c > 0.

Answer 1

Let f(x) = ax² + bx + c. Given ax2 + bx + c = 0 = f(x) = 0 has no real root, therefore, f(x) will have same sign for all real values of x. So,

⇒ f(0) and f(1) will have same sign

⇒ f(1)⋅f(0) > 0

⇒ (a + b + c) c > 0