Quadratic function

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,...