Any quadratic polynomial with two variables may be written as
:<math> f(x,y) = 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'', and ''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, whichas arethe characterized[[implicit equation]] of a conic section is obtained by equating theto expressionzero fora ''f''quadratic (''x''polynomial, ''y'')and tothe [[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 andor [[hypersurface]]s. In [[linear algebra]], quadratic polynomials can be generalized to the notion of a [[quadratic form]] on a [[vector space]].
Quadratic polynomials that have only terms of degree two are called [[quadratic form]]s.
