Editing Quadratic function (section)

==Terminology==

===Coefficients===
The [[coefficients]] of a quadratic function are often  taken to be [[real number|real]] or [[complex number]]s, but they may be taken in any [[ring (mathematics)|ring]], in which case the [[domain of a function|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 "[[Degeneracy (mathematics)|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 (mathematics)|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
:<math>ax^2 + bx + c,</math>
where ''x'' is the variable, and ''a'', ''b'', and ''c'' represent the [[coefficient]]s. Such polynomials often arise in a [[quadratic equation]] <math>ax^2 + bx + c = 0.</math>  The solutions to this equation are called the [[Root of a function|roots]] and can be expressed in terms of the coefficients as the [[quadratic formula]].  Each quadratic polynomial has an associated quadratic function, whose [[graph of a function|graph]] is a [[parabola]].

====Bivariate and multivariate cases====

Any quadratic polynomial with two variables may be written as
:<math>a x^2 + b y^2 + cxy + dx+ e y + f,</math>
where {{math|''x''}} and {{math|''y''}} are the variables and {{math|''a'', ''b'', ''c'', ''d'', ''e'', ''f''}} are the coefficients, and one of {{mvar|a}}, {{mvar|b}} and {{mvar|c}} is nonzero.  Such polynomials are fundamental to the study of [[conic section]]s, as the [[implicit equation]] of a conic section is obtained by equating to zero a quadratic polynomial, and the [[zero of a function|zeros]] of a quadratic function form a (possibly degenerate) conic section.

Similarly, quadratic polynomials with three or more variables correspond to [[quadric]] surfaces or [[hypersurface]]s. 

Quadratic polynomials that have only terms of degree two are called [[quadratic form]]s.