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Algebra Difficulty 4.9 AIME Prove it United States

Problem:

Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying
$(y+x) = (y-x)^2 + 3(y-x) + 3.$
Find the minimum possible value of $y$ .

Solution

Solution:

Let $w = y - x$ . Adding $w$ to both sides and dividing by two gives
$y = \frac{w^2 + 4w + 3}{2} = \frac{(w+2)^2 - 1}{2}$
which is minimized when $w = -2$ . This yields $y = -\frac{1}{2}$ .

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