Quadratic function

In mathematics, a quadratic function of a single variable is a function of the form

where ⁠${\displaystyle x}$⁠ is its variable, and ⁠${\displaystyle a}$⁠, ⁠${\displaystyle b}$⁠, and ⁠${\displaystyle c}$⁠ are coefficients. The expression ⁠${\displaystyle \textstyle ax^{2}+bx+c}$⁠, especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two. In elementary mathematics a polynomial and its associated polynomial function are rarely distinguished and the terms quadratic function and quadratic polynomial are nearly synonymous and often abbreviated as quadratic.

The graph of a real single-variable quadratic function is a parabola. If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula.

A quadratic polynomial or quadratic function can involve more than one variable. For example, a two-variable quadratic function of variables ⁠${\displaystyle x}$⁠ and ⁠${\displaystyle y}$⁠ has the form

with at least one of ⁠${\displaystyle a}$⁠, ⁠${\displaystyle b}$⁠, and ⁠${\displaystyle c}$⁠ not equal to zero. In general the zeros of such a quadratic function describe a conic section (a circle or other ellipse, a parabola, or a hyperbola) in the ⁠${\displaystyle x}$⁠–⁠${\displaystyle y}$⁠ plane. A quadratic function can have an arbitrarily large number of variables. The set of its zero form a quadric, which is a surface in the case of three variables and a hypersurface in general case.

The adjective quadratic comes from the Latin word quadrātum ("square"). A term raised to the second power like ⁠${\displaystyle \textstyle x^{2}}$⁠ is called a square in algebra because it is the area of a square with side ⁠${\displaystyle x}$⁠.

The coefficients of a quadratic function are often taken to be real or complex numbers, but they may be taken in any ring, in which case the domain and the codomain are this ring (see polynomial evaluation).

When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "degenerate case". Usually the context will establish which of the two is meant.