Problem
ALG-B1-M04-P013 Three Roots of a Product
#13
★★★★★ Level 5 of 5
Numbers \(u,v\) are such that each of \(x^2+ux+v\) and \(x^2+vx+u\) has two distinct roots, while their product has exactly three distinct roots. Find the sum of these three roots.
Inspired by final olympiad method · 2015 · Grade 9 · Problem 1
The trinomials have a common root. Subtract their values at that root.
Since the product has exactly three distinct roots, the trinomials have a common root \(r\). Also \(u\ne v\), otherwise there would be only two roots.
Subtract: \(0=(r^2+ur+v)-(r^2+vr+u)=(u-v)(r-1)\). Hence \(r=1\). Substitution gives \(1+u+v=0\).
Let the other roots be \(r_1,r_2\). By Vieta, \(1+r_1=-u\), \(1+r_2=-v\). Therefore the sum of the three distinct roots is \(1+r_1+r_2=1-u-v-2=-u-v-1=0\).
Preserves the common-root and Vieta idea; full direct factorisation is unnecessary.