In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".

For example, a univariate (single-variable) quadratic function has the form

${\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0,}$

where x is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the y-axis.

If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros of the corresponding quadratic function.

The bivariate case in terms of variables x and y has the form

${\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,}$

with at least one of a, b, c not equal to zero. The zeros of this quadratic function is, in general (that is, if a certain expression of the coefficients is not equal to zero), a conic section (a circle or other ellipse, a parabola, or a hyperbola).

A quadratic function in three variables x, y, and z contains exclusively terms x², y², z², xy, xz, yz, x, y, z, and a constant:

${\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,}$

where at least one of the coefficients a, b, c, d, e, f of the second-degree terms is not zero.

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.