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Problem 894

AMC 12 late, AIME early

Algebra Difficulty 4.7 Prove it Harvard-MIT Mathematics Tournament · United States

Find all irrational numbers $x$ such that $x^{3}-17 x$ and $x^{2}+4 x$ are both rational numbers.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:
Answer: $-2 \pm \sqrt{5}$
From $x^{2}+4 x \in \mathbb{Q}$ , we deduce that $(x+2)^{2}=x^{2}+4 x+4$ is also rational, and hence $x=-2 \pm \sqrt{y}$ , where $y$ is rational. Then $x^{3}-17 x=(26-6 y) \pm (y-5) \sqrt{y}$ , which forces $y$ to be $5$ . Hence $x=-2 \pm \sqrt{5}$ . It is easy to check that both values satisfy the problem conditions.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.