Completing the square
Named by 7 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
Never negative and no sum of squares
Any quadratic that never goes below nought can be completed into squares, and the squares are the proof. In 1888 Hilbert showed the same habit must fail at higher degree without exhibiting a single failure; the first one was written down 79 years later. Motzkin's x⁴y² + x²y⁴ − 3x²y² + 1 touches nought four times and never crosses it, and one cell of a four-by-four table shows that no squares can make it. Multiply it by 1 + x² + y² and five squares appear.
Named alongside it
The objects these essays reach for when they reach for this one.
Quadratic formNormal distributionQuadratic polynomialsConditional probabilityCorrelationProjectionRootsSimulationAlgebra tilesAreaBisectionCertificate