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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 6 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 One form completed three ways, and the same straddle each time. The plane coloured by the sign of 2x² + 6xy + 3y². Three pairs of lines, one for completing with x first, one for y first and one along the eigenvectors; their coefficients are 2 and −3/2; 3 and −1; 5.541 and −0.541, and every pair has one line in a positive wedge and one in a negative wedge.

The signs no completion can change

A quadratic in several variables can be completed square by square in many orders, and each order hands back different coefficients. What no order changes is how many come out positive and how many negative — and that count, read off a completion of A − tI, says how many eigenvalues lie below t without finding a single one.

algebra · Completing the square Every slice of a tilted bell is a bell, centred off the axis. Contour ellipses of a two-variable bell with correlation 0.6, five vertical slices drawn as bells, their centres on the line y = 0.6x, and the ellipses' long axis on the diagonal y = x.

Regression is a square completed halfway

A bell in two variables, sliced at a fixed x, is a bell in y — and completing the square in y alone says where it is centred: at ρx, on a line flatter than the ellipse's own axis. That gap is regression to the mean, and the same completion says it runs backwards in time just as well as forwards.

algebra · Completing the square The correlations that reverse a coefficient. rho = 0.6: wedge shares 0.2051 and 0.2051 of the ellipse; arcsin(0.6)/pi = 0.2048.

Holding one still turns the other over

A quantity can rise with an outcome and still predict it downwards once a second quantity is held fixed. Completing the square in two predictors says exactly when: the pair of correlations has to fall in a wedge whose angle is arcsin of the predictors' own correlation. Averaged over every possible correlation matrix the chance is exactly 1/4 − 1/π², about one coefficient in seven; two coefficients never reverse together, and with two dozen predictors almost two in five do.

algebra · Completing the square

Named alongside it

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

Normal distributionQuadratic formQuadratic polynomialsConditional probabilityCorrelationProjectionRootsSimulationAlgebra tilesAreaBisectionComplex numbers

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