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f (x) + cg (x) = 0 and f (x) + ch (x) = 0 have a common root, trinomials

Source: St. Petersburg 2016 9.1

May 1, 2019
algebratrinomialquadratic trinomialcommon real root

Problem Statement

Given three quadratic trinomials f,g,hf, g, h without roots. Their elder coefficients are the same, and all their coefficients for x are different. Prove that there is a number cc such that the equations f(x)+cg(x)=0f (x) + cg (x) = 0 and f(x)+ch(x)=0f (x) + ch (x) = 0 have a common root.
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