jmc

Question

Let be a complex number such that ^7 = 1 and 1. Let = + ^2 + ^4 and = ^3 + ^5 + ^6. Then and satisfy the quadratic x^2 + ax + b = 0for some real numbers a and b. Enter the ordered pair (a,b).

Accepted Answer

From the equation ^7 = 1, ^7 - 1 = 0, which factors as ( - 1)(^6 + ^5 + ^4 + ^3 + ^2 + + 1) = 0.Since 1, ^6 + ^5 + ^4 + ^3 + ^2 + + 1 = 0.We have that + = + ^2 + ^4 + ^3 + ^5 + ^6 = -1.Also, align* &= ( + ^2 + ^4)(^3 + ^5 + ^6) &= ^4 + ^6 + ^7 + ^5 + ^7 + ^8 + ^7 + ^9 + ^10 &= ^4 + ^6 + 1 + ^5 + 1 + + 1 + ^2 + ^3 &= 2 + (^6 + ^5 + ^4 + ^3 + ^2 + + 1) &= 2. align*Then by Vieta's formulas, and are the roots of x^2 + x + 2 = 0, so (a,b) = (1,2).