MathDB

14

Part of 1969 IMO Shortlist

Problems(1)

Quadratic equation and proof of existence

Source:

9/29/2010
(CZS3)(CZS 3) Let aa and bb be two positive real numbers. If xx is a real solution of the equation x2+px+q=0x^2 + px + q = 0 with real coefficients pp and qq such that ∣p∣≤a,∣q∣≤b,|p| \le a, |q| \le b, prove that ∣x∣≤12(a+a2+4b)|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b}) Conversely, if xx satisfies the above inequality, prove that there exist real numbers pp and qq with ∣p∣≤a,∣q∣≤b|p|\le a, |q|\le b such that xx is one of the roots of the equation x2+px+q=0.x^2+px+ q = 0.
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