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Quadratic polynomials with repeated roots

Source: 42nd International Tournament of Towns, Senior O-Level P1, Fall 2020

February 18, 2023

algebrapolynomialTournament of Towns

Problem Statement

Each of the quadratic polynomials

P(x), Q(x)

and

P(x)+Q(x)

with real coefficients has a repeated root. Is it guaranteed that those roots coincide?
Boris Frenkin