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]