If α and β are the distinct roots of the equation x 2 + 3 1 4 x + 3 1 2 = 0 , then the value of α 96 α 12 - 1 + β 96 β 12 - 1 is equal to:

Step-by-step solution

We have, x 2 + 3 1 4 x + 3 1 2 = 0 Since, α is the root, therefore α 2 + 3 = - 3 1 4 · α On squaring, both sides we get α 4 + 2 3 α 2 + 3 = 3 α 2 ⇒ α 4 + 3 = - 3 α 2 On squaring, both sides we get α 8 + 6 α 4 + 9 = 3 α 4 ⇒ α 8 = - 9 - 3 α 4 Multiply by α 4 , we get α 12 = - 9 α 4 - 3 α 8 ∴   α 12 = - 9 α 4 - 3 - 9 - 3 α 4 ⇒ α 12 = 27 ⇒ α 12 8 = 27 8 ⇒ α 96