Quadratic equation

Quadratic equation

General form

$x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0.$

Derivation of the formula

The quadratic formula can be derived with a simple application of technique of completing the square.Divide the quadratic equation by a, which is allowed because a is non-zero:

$x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0.$

Subtract c/a from both sides of the equation, yielding:

$x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}.$

The quadratic equation is now in a form to which the method of completing the square can be applied. Thus, add a constant to both sides of the equation such that the left hand side becomes a complete square:

$x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2},$

which produces:

$\left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}.$

Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:

$\left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.$

The square has thus been completed. Taking the square root of both sides yields the following equation:

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

Isolating x gives the quadratic formula:

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

The plus-minus symbol "±" indicates that both

$x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}$

are solutions of the quadratic equation. There are many alternatives of this derivation with minor differences, mostly concerning the manipulation of a.

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax² − 2bx + c = 0 or ax² + 2bx + c = 0, where b has a magnitude one half of the more common one. These result in slightly different forms for the solution, but are otherwise equivalent.

A lesser known quadratic formula, as used in Muller's method, and which can be found from Vieta's formulas, provides the same roots via the equation:

$x={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac}}}}.$