Let a ∈ R and let α , β be the roots of the equation x 2 + 60 1 4 x + a = 0 . If α 4 + β 4 = - 30 , then the product of all possible values of a is _____ .

Step-by-step solution

Since, α , β be the roots of the equation x 2 + 60 1 4 x + a = 0 , therefore α + β = - 60 1 4 α β = a Given: α 4 + β 4 = - 30 ⇒ α 2 + β 2 2 - 2 α 2 β 2 = - 30 ⇒ α + β 2 - 2 α β 2 - 2 a 2 = - 30 ⇒ 60 1 2 - 2 a 2 - 2 a 2 = - 30 ⇒   60 + 4 a 2 - 4 a × 60 1 2 - 2 a 2 = - 30 ⇒   2 a 2 - 4 · 60 1 2 a + 90 = 0 ⇒ a 2 - 2 · 60 1 2 a + 45 = 0 Product of roots = 45