Let and be the roots of the quadratic equation x^2 - x + 2 = 0 , where R . If ^4 + ^4 = 17 , then the value of ^6 + ^6 is equal to

Step-by-step solution

Let the roots of the equation x^2 - x + 2 = 0 be and . From Vieta's formulas, we have: + = = 2 We are given that ^4 + ^4 = 17 . Using the identity ^4 + ^4 = ( ^2 + ^2)^2 - 2 ^2 ^2 , we get: 17 = ( ^2 + ^2)^2 - 2(2)^2 17 = ( ^2 + ^2)^2 - 8 ( ^2 + ^2)^2 = 25 This gives ^2 + ^2 = 5 or ^2 + ^2 = -5 . Since ^2 + ^2 = ( + )^2 - 2 = ^2 - 4 , and is a real number, we must have ^2 0 , which implies ^2 + ^2 -4 . Therefore, we reject -5 and accept ^2 + ^2 = 5 . Now, we need to find the value of ^6 + ^6 . Using the sum of cube