Quadratic formula

In <a href="/facts/Elementary_algebra/973nufT7">elementary algebra</a>, the quadratic formula is a <a href="/facts/Closed-form_expression/y0xCXlSk">closed-form expression</a> describing the solutions of a <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equation</a>. Other ways of solving quadratic equations, such as <a href="/facts/Completing_the_square/SN6xA8sc">completing the square</a>, 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 <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> ⁠
 
 
 
 a
 
 
 {\displaystyle a}
 
⁠, ⁠
 
 
 
 b
 
 
 {\displaystyle b}
 
⁠, and ⁠
 
 
 
 c
 
 
 {\displaystyle c}
 
⁠ representing known <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> numbers with ⁠
 
 
 
 a
 ≠
 0
 
 
 {\displaystyle a\neq 0}
 
⁠, the values of ⁠
 
 
 
 x
 
 
 {\displaystyle x}
 
⁠ satisfying the equation, called the <a href="/facts/Zero_of_a_function/mlK3dhoT">roots</a> 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 <a href="/facts/Plus%25E2%2580%2593minus_sign/cAP8AgPH">plus–minus symbol</a> "⁠
 
 
 
 ±
 
 
 {\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 <a href="/facts/Discriminant/0SmQtrvU">discriminant</a> 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 <a href="/facts/Real_number/R02gw5Pb">real</a> roots; when ⁠
 
 
 
 Δ
 =
 0
 
 
 {\displaystyle \Delta =0}
 
⁠, the equation has one <a href="/facts/Repeated_root/dndUyetA">repeated</a> real root; and when ⁠
 
 
 
 Δ
 <
 0
 
 
 {\displaystyle \Delta <0}
 
⁠, the equation has no real roots but has two distinct complex roots, which are <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugates</a> of each other.
Geometrically, the roots represent the ⁠
 
 
 
 x
 
 
 {\displaystyle x}
 
⁠ values at which the <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of the <a href="/facts/Quadratic_function/mKxHIC3U">quadratic function</a> ⁠
 
 
 
 
 y
 =
 a
 
 x
 
 2
 
 
 +
 b
 x
 +
 c
 
 
 
 {\displaystyle \textstyle y=ax^{2}+bx+c}
 
⁠, a <a href="/facts/Parabola/Cweo3wgw">parabola</a>, 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 <a href="/facts/Axis_of_symmetry/bpvJqcnS">axis of symmetry</a>.

Quadratic formula open-in-new

Quadratic formula