Portal:Mathematics/Selected article/31

The graph of a real-valued quadratic function of a real variable x, is a parabola. Image credit: Enoch Lau

A quadratic equation is a polynomial equation of degree two. The general form is

${\displaystyle ax^{2}+bx+c=0,\,\!}$

where a ≠ 0 (if a = 0, then the equation becomes a linear equation). The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x², the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term.

Quadratic equations are known by that name because quadratus is Latin for "square"; in the leading term the variable is squared.

A quadratic equation has two (not necessarily distinct) solutions, which may be real or complex, given by the quadratic formula:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},}$

If the discriminant ${\displaystyle b^{2}-4ac>0}$, then the quadratic equation has two distinct real solutions; if ${\displaystyle b^{2}-4ac=0}$, the equation has two real solutions which are equal; if ${\displaystyle b^{2}-4ac<0}$, the equation has two complex solutions.

These solutions are roots of the corresponding quadratic function

${\displaystyle f(x)=ax^{2}+bx+c.\,}$ (Full article...)