Estonija, 2010

Algebra Difficulty 4.9 AIME Prove it Estonia

Let $a$ be a fixed real number. Find all real numbers $b$ such that, for every real number $x$ , at least one of the numbers $x^2 + ax + b$ and $x^2 - ax + b$ is non-negative.

Solution

Note that $x^2 + ax + b$ and $x^2 - ax + b$ sum up to $2x^2 + 2b$ . If $b \ge 0$ , then it is non-negative for arbitrary real number $x$ , implying that at least one of the numbers added was non-negative. If $b < 0$ , then taking $x = 0$ turns both summands negative.

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