Question

Find the values of a for which all roots of the equation 3x4 + 4x3 – 12x2 + a = 0 are real and distinct.

Answer 1

Let f(x) = 3x4 + 4x3 – 12x2 + a

⇒ f'(x) = 12x3 + 12x2 – 24x

= 12x(x2 + x – 2)

= 12x(x – 1)(x + 2)

The function f(x) will provide us four real and distinct roots if f(– 2) < 0, f(0) > 0, f(1) < 0

⇒ 48 – 32 – 48 + a < 0, a > 0, 3 + 4 – 12 + a < 0

⇒ a < 32, a > 0, a < 5

⇒ 0 < a < 5

⇒ a ∈ (0, 5)