jmc

Question

Let a and b be real numbers such that the quadratic equations x^2 + ax + b = 0 and ax^2 + bx + 1 = 0 have a root in common. Enter all possible values of a + b, separated by commas.

Accepted Answer

Let r be the common root, so align* r^2 + ar + b &= 0, ar^2 + br + 1 &= 0. align*Then r^3 + ar^2 + br = 0, so r^3 = 1. Then r^3 - 1 = 0, which factors as (r - 1)(r^2 + r + 1) = 0. If r = 1, then 1 + a + b = 0, so a + b = -1. If r^2 + r + 1 = 0, then r is nonreal, so we must have a = b = 1. Thus, the only possible values of a + b are -1,2.