Quadratic Formula - Wikipedia

Not to be confused with quadratic function or quadratic equation.

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.[1] 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.[2] 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.[3] The quadratic formula can also be used to identify the parabola's axis of symmetry.[4]

Contents

• 1 Derivation by completing the square
• 2 Equivalent formulations
  • 2.1 Square root in the denominator
• 3 Other derivations
  • 3.1 Completing the square by Śrīdhara's method
  • 3.2 By substitution
  • 3.3 By using algebraic identities
  • 3.4 By Lagrange resolvents
• 4 Numerical calculation
• 5 Historical development
• 6 Geometric significance
• 7 Dimensional analysis
• 8 See also
• 9 Notes
• 10 References