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An Equation Satisfied by the Coefficients of Polynomials

Source: Czech-Polish-Slovak Match, 2004

September 13, 2012
algebrapolynomialquadraticsalgebra unsolved

Problem Statement

Show that real numbers, p,q,rp, q, r satisfy the condition p4(q−r)2+2p2(q+r)+1=p4p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4 if and only if the quadratic equations x2+px+q=0x^2 + px + q = 0 and y2−py+r=0y^2 - py + r = 0 have real roots (not necessarily distinct) which can be labeled by x1,x2x_1,x_2 and y1,y2y_1,y_2, respectively, in such a way that x1y1−x2y2=1x_1y_1 - x_2y_2 = 1.
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