Quadratic formula

In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions.

Given a general quadratic equation of the form ⁠${\displaystyle \textstyle ax^{2}+bx+c=0}$⁠, with ⁠${\displaystyle x}$⁠ representing an unknown, and coefficients ⁠${\displaystyle a}$⁠, ⁠${\displaystyle b}$⁠, and ⁠${\displaystyle c}$⁠ representing known real or complex numbers with ⁠${\displaystyle a\neq 0}$⁠, the values of ⁠${\displaystyle x}$⁠ satisfying the equation, called the roots or zeros, can be found using the quadratic formula,

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

where the plus–minus symbol "⁠${\displaystyle \pm }$⁠" indicates that the equation has two roots.^[1] Written separately, these are:

$${\displaystyle x_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}},\qquad x_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}.}$$

The quantity ⁠${\displaystyle \textstyle \Delta =b^{2}-4ac}$⁠ is known as the discriminant of the quadratic equation.^[2] If the coefficients ⁠${\displaystyle a}$⁠, ⁠${\displaystyle b}$⁠, and ⁠${\displaystyle c}$⁠ are real numbers then when ⁠${\displaystyle \Delta >0}$⁠, the equation has two distinct real roots; when ⁠${\displaystyle \Delta =0}$⁠, the equation has one repeated real root; and when ⁠${\displaystyle \Delta <0}$⁠, the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other.

Geometrically, the roots represent the ⁠${\displaystyle x}$⁠ values at which the graph of the quadratic function ⁠${\displaystyle \textstyle y=ax^{2}+bx+c}$⁠, a parabola, crosses the ⁠${\displaystyle x}$⁠-axis: the graph's ⁠${\displaystyle x}$⁠-intercepts.^[3] The quadratic formula can also be used to identify the parabola's axis of symmetry.^[4]