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2017 T10: Vieta Quadratic Interpolation

Source:

January 29, 2017
algebrapolynomialVieta

Problem Statement

The polynomial P(x)=x3−6x−2P(x) = x^3 - 6x - 2 has three real roots, α\alpha, β\beta, and γ\gamma. Depending on the assignment of the roots, there exist two different quadratics QQ such that the graph of y=Q(x)y=Q(x) pass through the points (α,β)(\alpha,\beta), (β,γ)(\beta,\gamma), and (γ,α)(\gamma,\alpha). What is the larger of the two values of Q(1)Q(1)?
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