Quadratic formula

In elementary algebra, the quadratic formula is a formula that provides the two solutions, or roots, to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as completing the square.

Given a general quadratic equation of the form

ax^{2}+bx+c=0

with $x$ representing an unknown, with $a$ , $b$ and $c$ representing constants, and with $a \neq 0$ , the quadratic formula is:

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

where the plus–minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become:

{\begin{aligned}x_{1}&={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\\x_{2}&={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\end{aligned}}

Each of these two solutions is also called a root (or zero) of the quadratic equation. Geometrically, these roots represent the $x$ -values at which any parabola, explicitly given as $y = ax 2 + bx + c$ , crosses the $x$ -axis.

As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola, and the number of real zeros the quadratic equation contains.

The expression $Δ = b 2 - 4 ac$ is known as the discriminant. If $a$ , $b$ , and $c$ are real numbers and $a \neq 0$ then

When $b 2 - 4 ac > 0$ , there are two distinct real roots or solutions to the equation $ax 2 + bx + c = 0$ .
When $b 2 - 4 ac = 0$ , there is one repeated real solution.
When $b 2 - 4 ac < 0$ , there are two distinct complex solutions, which are complex conjugates of each other.

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