MathDB

5

Part of 2005 Czech And Slovak Olympiad III A

Problems(1)

x^2 + px+q = 0, x^2 +rx+s = 0 => pr = (q+1)(s+1) , p(q+1)s = r(s+1)q

Source: Czech and Slovak MO, III A, 2005 p5

1/12/2020
Let p,q,r,sp,q, r, s be real numbers with q≠−1q \ne -1 and s≠−1s \ne -1. Prove that the quadratic equations x2+px+q=0x^2 + px+q = 0 and x2+rx+s=0x^2 +rx+s = 0 have a common root, while their other roots are inverse of each other, if and only if pr=(q+1)(s+1)pr = (q+1)(s+1) and p(q+1)s=r(s+1)qp(q+1)s = r(s+1)q. (A double root is counted twice.)
quadraticsquadratic trinomialtrinomialalgebra