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Quadratic Formula ○◂｜Definition｜1st｜20251119205401-00-⌔

Quadratic formula - Wikipedia

Quadratic formula

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

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

where the plus–minus symbol ”⁠ $\pm$ ⁠” indicates that the equation has two roots.¹ Written separately, these are:

$x_1=\frac{-b+\sqrt{b^2-4ac}}{2a},x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}.$

The quantity ⁠ $\Delta =b^2-4ac$ ⁠ is known as the discriminant of the quadratic equation.² If the coefficients ⁠ $a$ ⁠, ⁠ $b$ ⁠, and ⁠ $c$ ⁠ are real numbers then when ⁠ $\Delta >0$ ⁠, the equation has two distinct real roots; when ⁠ $\Delta =0$ ⁠, the equation has one repeated real root; and when ⁠ $\Delta <0$ ⁠, the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other.

Geometrically, the roots represent the ⁠ $x$ ⁠ values at which the graph of the quadratic function ⁠ $y=a{x}^2+bx+c$ ⁠, a parabola, crosses the ⁠ $x$ ⁠ -axis: the graph’s ⁠ $x$ ⁠ -intercepts.³ The quadratic formula can also be used to identify the parabola’s axis of symmetry.⁴

Printed 2026-06-28.

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Footnotes

1. Sterling, Mary Jane (2010), Algebra I For Dummies, Wiley Publishing, p. 219, ISBN 978-0-470-55964-2 ↩

2. “Discriminant review”, Khan Academy, retrieved 2019-11-10 ↩

3. “Understanding the quadratic formula”, Khan Academy, retrieved 2019-11-10 ↩

4. “Axis of Symmetry of a Parabola. How to find axis from equation or from a graph. To find the axis of symmetry…”, www.mathwarehouse.com, retrieved 2019-11-10 ↩

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