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

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

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

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

The quantity Δ = b 2 − 4 a c {\displaystyle \textstyle \Delta =b^{2}-4ac} is known as the discriminant of the quadratic equation. If the coefficients a {\displaystyle a}, b {\displaystyle b}, and c {\displaystyle c} are real numbers then when Δ > 0 {\displaystyle \Delta >0}, the equation has two distinct real roots; when Δ = 0 {\displaystyle \Delta =0}, the equation has one repeated real root; and when Δ < 0 {\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 x {\displaystyle x} values at which the graph of the quadratic function y = a x 2 + b x + c {\displaystyle \textstyle y=ax^{2}+bx+c}, a parabola, crosses the x {\displaystyle x} -axis: the graph's x {\displaystyle x} -intercepts. The quadratic formula can also be used to identify the parabola's axis of symmetry.
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