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x^3+\sqrt{3}(a-1)x^2-6ax+b=0

Source: Argentina 1995 OMA L3 p5

May 13, 2024
algebra

Problem Statement

Let a,ba,b be real numbers such that the equation x3+3(a−1)x2−6ax+b=0x^3+\sqrt{3}(a-1)x^2-6ax+b=0has three real roots. Prove that ∣b∣≤∣a+1∣3|b|\leq |a+1|^3.
>Clarification: ∣x∣|x| indicates the absolute value of xx. For example, ∣5∣=5|5|=5; ∣−1.23∣=1.23|-1.23|=1.23; etc
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