Editing Quadratic formula (section)

{{Short description|Formula that provides the solutions to a quadratic equation}}
{{confused|quadratic function|quadratic equation}}
[[File:Roots of a quadratic function via the quadratic formula.png|alt=A graph of a parabola-shaped function which intersects the x-axis at x = 1 and x = 5|thumb|The roots of the quadratic function {{math|1=''y'' = {{sfrac|1|2}}''x''<sup>2</sup> − 3''x'' + {{sfrac|5|2}}}} are the places where the graph intersects the {{mvar|x}}-axis, the values {{math|1=''x'' = 1}} and {{math|1=''x'' = 5}}. They can be found via the 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 {{tmath|1=\textstyle ax^2 + bx + c = 0}}, with {{tmath|x}} representing an unknown, and [[coefficient]]s {{tmath|a}}, {{tmath|b}}, and {{tmath|c}} representing known [[real number|real]] or [[complex number|complex]] numbers with {{tmath|a \neq 0}}, the values of {{tmath|x}} satisfying the equation, called the [[Zero of a function|''roots'']] or ''zeros'', can be found using the quadratic formula,

<math display=block>
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
</math>

where the [[plus–minus sign|plus–minus symbol]] "{{tmath|\pm}}" indicates that the equation has two roots.<ref>{{Citation|last=Sterling|first=Mary Jane|title=Algebra I For Dummies | year=2010 | publisher=Wiley Publishing | isbn=978-0-470-55964-2 | url=https://books.google.com/books?id=2toggaqJMzEC&q=quadratic+formula&pg=PA219 | page=219}}</ref> Written separately, these are:

<math display=block>
x_1 = \frac{-b + \sqrt {b^2 - 4ac}}{2a}, \qquad
x_2 = \frac{-b - \sqrt {b^2 - 4ac}}{2a}.
</math>

The quantity {{tmath|1=\textstyle \Delta = b^2 - 4ac}} is known as the [[discriminant]] of the quadratic equation.<ref>{{Cite web |mode=cs2 | url=https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-formula-a1/a/discriminant-review|title=Discriminant review|website=Khan Academy|language=en|access-date=2019-11-10}}</ref> If the coefficients {{tmath|a}}, {{tmath|b}}, and {{tmath|c}} are real numbers then when {{tmath|\Delta > 0}}, the equation has two distinct [[real number|real]] roots; when {{tmath|1= \Delta = 0}}, the equation has one [[repeated root|repeated]] real root; and when {{tmath|\Delta < 0}}, the equation has ''no'' real roots but has two distinct complex roots, which are [[complex conjugate]]s of each other.

Geometrically, the roots represent the {{tmath|x}} values at which the [[graph of a function|graph]] of the [[quadratic function]] {{tmath|1=\textstyle y = ax^2 + bx + c}}, a [[parabola]], crosses the {{tmath|x}}-axis: the graph's {{tmath|x}}-intercepts.<ref>{{Cite web |mode=cs2 |url=https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-formula-a1/a/quadratic-formula-explained-article| title=Understanding the quadratic formula|website=Khan Academy|language=en|access-date=2019-11-10}}</ref> The quadratic formula can also be used to identify the parabola's [[axis of symmetry]].<ref>{{Cite web |mode=cs2 |url=https://www.mathwarehouse.com/geometry/parabola/axis-of-symmetry.php|title=Axis of Symmetry of a Parabola. How to find axis from equation or from a graph. To find the axis of symmetry ...|website=www.mathwarehouse.com|access-date=2019-11-10}}</ref>