Completing the square

Completing the square rewrites a quadratic expression as a perfect square plus a constant. For a monic quadratic x² + bx, add and subtract (b/2)² to obtain (x + b/2)² − (b/2)². For the general form ax² + bx + c, first factor out a, then complete the square inside the parentheses, yielding a(x + b/(2a))² + (c − b²/(4a)). This form reveals the vertex of the parabola: the minimum or maximum occurs at x = −b/(2a).¹

Completing the square is essential for solving quadratic equations when factoring is not straightforward. By isolating the perfect square and taking the square root, one obtains the quadratic formula x = (−b ± √(b² − 4ac))/(2a).² It also simplifies integrals involving quadratics, such as ∫ dx/(x² + px + q), by transforming the denominator into a sum of squares, a technique frequently used in calculus and differential equations.³

The method predates algebraic notation, appearing in geometric form in ancient Babylonian and Greek mathematics. Al-Khwarizmi's 9th-century treatise Al-Jabr solved quadratic equations by literally completing a square geometrically, adding a small square to a larger one to find the side length.⁴ The algebraic formulation as we know it emerged in the 16th century with European mathematicians like Michael Stifel and François Viète, who introduced symbolic manipulation of variables.⁵

Beyond quadratics, completing the square generalizes to higher-degree polynomials and matrix equations. In linear algebra, it underlies the Cholesky decomposition and the method of least squares.⁶ In number theory, it is used to prove that every positive integer is a sum of four squares (Lagrange's theorem) by completing the square in the context of quaternions.⁷ The technique also appears in statistics when deriving the normal distribution's density, and in physics when analyzing harmonic oscillators. A curious edge case: when the coefficient a is negative, completing the square still works but the vertex represents a maximum rather than a minimum.⁸

Perfect square trinomial

A trinomial of the form x² + 2px + p², which factors as (x + p)².

Quadratic formula

The formula x = (−b ± √(b² − 4ac))/(2a) for solving ax² + bx + c = 0.

Vertex

The point on a parabola where it changes direction; its x-coordinate is −b/(2a).