Editing Quadratic equation (section)

=== Quadratic formula and its derivation ===
{{Main|Quadratic formula}}
[[Completing the square]] can be used to [[Quadratic formula#Derivations|derive a general formula]] for solving quadratic equations, called the quadratic formula.<ref>{{citation
|title=Schaum's Outline of Theory and Problems of Elementary Algebra
|first1=Barnett
|last1=Rich
|first2=Philip
|last2=Schmidt
|publisher=The McGraw-Hill Companies
|year=2004
|isbn=978-0-07-141083-0
|url=https://books.google.com/books?id=8PRU9cTKprsC}}, [https://books.google.com/books?id=8PRU9cTKprsC&pg=PA291 Chapter 13 §4.4, p. 291]</ref> The [[mathematical proof]] will now be briefly summarized.<ref>Himonas, Alex.  ''[https://books.google.com/books?id=1Mg5u98BnEMC&q=%22left+as+an+exercise%22+and+%22quadratic+formula%22 Calculus for Business and Social Sciences]'', p. 64 (Richard Dennis Publications, 2001).</ref>  It can easily be seen, by [[polynomial expansion]], that the following equation is equivalent to the quadratic equation:
<math display="block">\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.</math>
Taking the [[square root]] of both sides, and isolating {{math|''x''}}, gives:
<math display="block">x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.</math>

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as {{math|''ax''<sup>2</sup> + 2''bx'' + ''c'' {{=}} 0}} or {{math|''ax''<sup>2</sup> &minus; 2''bx'' + ''c'' {{=}} 0}}&nbsp;,<ref name="kahan">{{Citation |first=Willian |last=Kahan |title=On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic |url=http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf |date=November 20, 2004 |access-date=2012-12-25}}</ref> where {{math|''b''}} has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.

A number of [[Quadratic formula#Other derivations|alternative derivations]] can be found in the literature.  These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.

A lesser known quadratic formula, as used in [[Muller's method]], provides the same roots via the equation
<math display="block">x = \frac{2c}{-b \pm \sqrt {b^2-4ac}}.</math> 
This can be deduced from the standard quadratic formula by [[Vieta's formulas]], which assert that the product of the roots is {{math|''c''/''a''}}. It also follows from dividing the quadratic equation by <math>x^2</math> giving <math>cx^{-2}+bx^{-1}+a=0,</math> solving this for <math>x^{-1},</math> and then inverting.

One property of this form is that it yields one valid root when {{math|''a'' {{=}} 0}}, while the other root contains division by zero, because when {{math|''a'' {{=}} 0}}, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an [[indeterminate form]] {{math|0/0}} for the other root. On the other hand, when {{math|''c'' {{=}} 0}}, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form {{math|0/0}}.

When neither {{mvar|a}} nor {{mvar|c}} is zero, the equality between the standard quadratic formula and Muller's method,
<math display="block">\frac{2c}{-b - \sqrt {b^2-4ac}} = \frac{-b + \sqrt {b^2-4ac}}{2a}\,,</math>
can be verified by [[cross multiplication]], and similarly for the other choice of signs.