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Concept

Completing the square

Rewriting a quadratic as a squared term plus a constant, so that x squared plus bx becomes the square of x plus b/2, less b squared over four. It turns a quadratic into a shifted square, which gives the quadratic formula and locates the vertex of a parabola.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Completing the square, as a square. An x by x square with the strip split in half and laid along two sides, leaving a square hole of side 1.5. Filling the hole costs 2.25 and buys a perfect square.

Completing the square, by completing a square

The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.

algebra · Completing the square Every quadratic is a point. The plane of monic quadratics x² + px + q with p across and q up. The parabola q = p²/4 divides it: the region below, shaded, holds the equations with two real roots, the curve itself the ones with a repeated root, and the region above the ones with none. 5 equations are marked and labelled.

Where two roots run into each other

Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.

algebra · Completing the square A quadratic in the exponent, completed. Two panels sharing an x-axis. Above, the parabola −x² + 2x with its top at x = 1 marked. Below, e raised to that parabola: a bell centred at the same x = 1, with peak height e^1, beside the faint unmoved bell e^(−x²).

One number under every bell

The area under e^(−x²) has no formula in terms of the usual functions, and yet the area under e raised to any downward quadratic is known exactly. Completing the square in the exponent moves and squeezes every such curve into the same one, so a single number — √π — pays for all of them.

algebra · Completing the square

Named alongside it

The objects these essays reach for when they reach for this one.

Quadratic polynomialsRootsAlgebra tilesAreaComplex numbersCubicDiscriminantDissectionDualitye, the numberFourier analysisImaginary unit

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