Let α , β be the roots of the equation x 2 - x + 2 = 0 with Im ( α ) > Im ( β ) . Then α 6 + α 4 + β 4 - 5 α 2 is equal to

Step-by-step solution

Given equation x 2 - x + 2 has roots α & β , So, α + β = 1 , α β = 2 ⇒ α 4 + β 4 = α 2 + β 2 2 - 2 α 2 β 2 ⇒ α 4 + β 4 = α + β 2 - 2 α β 2 - 2 α 2 β 2 ⇒ α 4 + β 4 = 1 - 4 2 - 2 × 4 ⇒ α 4 + β 4 = 1 It is given that, α is a root of x 2 - x + 2 . ⇒ α 2 - α + 2 = 0 ⇒ α 2 = α - 2 ⇒ α 4 = α 2 + 4 - 4 α ⇒ α 4 = α - 2 + 4 - 4 α ⇒ α 4 = 2 - 3 α Now, α 6 - 5 α 2 = α 2 α 4 - 5 ⇒ α 6 - 5 α 2 = α - 2 2 - 3 α - 5 ⇒ α 6 - 5 α 2 = α - 2 - 3 α - 3 ⇒ α 6 - 5 α 2 = - 3 α 2 - α - 2 ⇒ α 6 - 5 α 2 = - 3 α - 2 - α - 2 ⇒ α 6 - 5 α 2 = 12