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]

Derivation by completing the square

The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation ${\displaystyle \textstyle ax^{2}+bx+c=0}$.^[5][6][7][8] The idea is to manipulate the equation into the form ${\displaystyle \textstyle (x+k)^{2}=s}$ for some expressions ${\displaystyle k}$ and ${\displaystyle s}$ written in terms of the coefficients; take the square root of both sides; and then isolate ${\displaystyle x}$.

We start by dividing the equation by the quadratic coefficient ${\displaystyle a}$, which is allowed because ${\displaystyle a}$ is non-zero. Afterwards, we subtract the constant term ${\displaystyle c/a}$ to isolate it on the right-hand side:

$${\displaystyle {\begin{aligned}ax^{2{\vphantom {|}}}+bx+c&=0\\x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}&=0\\x^{2}+{\frac {b}{a}}x&=-{\frac {c}{a}}.\end{aligned}}}$$

The left-hand side is now of the form ${\displaystyle \textstyle x^{2}+2kx}$, and we can "complete the square" by adding a constant ${\displaystyle \textstyle k^{2}}$ to obtain a squared binomial ${\displaystyle \textstyle x^{2}+2kx+k^{2}={}}$​${\displaystyle \textstyle (x+k)^{2}}$. In this example we add ${\displaystyle \textstyle (b/2a)^{2}}$ to both sides so that the left-hand side can be factored: