In mathematics, a quadratic function of a single variable is a function of the form
f ( x ) = a x 2 + b x + c {\displaystyle f(x)=ax^{2}+bx+c}
with a ≠ 0 {\displaystyle a\neq 0} , where x {\displaystyle x} is its variable, and a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are coefficients. The expression a x 2 + b x + c {\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 x {\displaystyle x} and y {\displaystyle y} has the form
f ( x , y ) = a x 2 + b x y + c y 2 + d x + e y + f , {\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,}
with at least one of a {\displaystyle a} , b {\displaystyle b} , and c {\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 x {\displaystyle x} – y {\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.
Etymology The adjective quadratic comes from the Latin word quadrātum ("square"). A term raised to the second power like x 2 {\displaystyle \textstyle x^{2}} is called a square in algebra because it is the area of a square with side x {\displaystyle x} .
Terminology
Coefficients 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).
Degree 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. Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial. However, where the "degree of a polynomial" refers to the largest degree of a non-zero term of the polynomial, more typically "order" refers to the lowest degree of a non-zero term of a power series.
Variables A quadratic polynomial may involve a single variable x (the univariate case), or multiple variables such as x, y, and z (the multivariate case).
The one-variable case Any single-variable quadratic polynomial may be written as
a x 2 + b x + c , {\displaystyle ax^{2}+bx+c,}
where x is the variable, and a, b, and c represent the coefficients. Such polynomials often arise in a quadratic equation a x 2 + b x + c = 0. {\displaystyle ax^{2}+bx+c=0.} The solutions to this equation are called the roots and can be expressed in terms of the coefficients as the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.
Bivariate and multivariate cases Any quadratic polynomial with two variables may be written as
a x 2 + b y 2 + c x y + d x + e y + f , {\displaystyle ax^{2}+by^{2}+cxy+dx+ey+f,}
where x and y are the variables and a, b, c, d, e, f are the coefficients, and one of a, b and c is nonzero. Such polynomials are fundamental to the study of conic sections, as the implicit equation of a conic section is obtained by equating to zero a quadratic polynomial, and the zeros of a quadratic function form a (possibly degenerate) conic section. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces or hypersurfaces. Quadratic polynomials that have only terms of degree two are called quadratic forms.
Forms of a univariate quadratic function A univariate quadratic function can be expressed in three formats:
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