jmc

Question

Let a and b be the positive roots of x^2 - 3x + 1 = 0. Find ab + ba.

Accepted Answer

By Vieta's formulas, a + b = 3 and ab = 1. Let t = ab + ba.Then align* t^2 &= a^2b + 2 ab + b^2a &= a^3 + b^3ab + 2 &= (a + b)(a^2 - ab + b^2)ab + 2 &= (a + b)((a + b)^2 - 3ab)ab + 2 &= 3 (3^2 - 3)1 + 2 &= 20, align*so t = 20 = 2 5.