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Completing the Square

This is just a small observation that serves to illustrate the sort of thing we attempt to promote in Calculus&Mathematica courses. It's not from the course, but the spirit is there.

A quadratic expression like x² + A x + B (with real coefficients A and B) can always be put into the form (x + a)² + b. The latter form is good for examining numbers and locations of zeros of the expression (even the quadratic formula), and is useful in performing other calculations like integrals which include terms or factors like x² + A x + B.

Given A and B, and knowing your goal, namely finding a and b, how would you proceed? I know I'd try expanding things and solving for a and b. The important thing to me would be to realize that it's possible and useful.

Ask pre-calculus students about completing the square. I'll bet they can tell you what a and b are from their memorized formula. I'll also bet they don't have good reasons for knowing why you'd do something like that.

What's the point? Which would you rather have your students know? Which are they more likely to recall?