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Question

The roots of x^4 - Kx^3 + Kx^2 + Lx + M = 0 are a, b, c, and d. If K, L, and M are real numbers, compute the minimum value of the sum a^2 + b^2 + c^2 + d^2.

Accepted Answer

By Vieta's formulas, a + b + c + d = K and ab + ac + ad + bc + bd + cd = K. Squaring the equation a + b + c + d = K, we get a^2 + b^2 + c^2 + d^2 + 2(ab + ac + ad + bc + bd + cd) = K^2.Hence, a^2 + b^2 + c^2 + d^2 = K^2 - 2K = (K - 1)^2 - 1.This expression is minimized at K = 1, with a minimum value of -1.