Editing Quadratic formula (section)

== Equivalent formulations ==
The quadratic formula can equivalently be written using various alternative expressions, for instance

<math display=block>
x = -\frac{b}{2a} \pm \sqrt{\left(\frac{b}{2a}\right)^2-\frac{c}{a}},
</math>

which can be derived by first dividing a quadratic equation by {{tmath|2a}}, resulting in {{tmath|1=\textstyle \tfrac12 x^2 + \tfrac{b}{2a}x + \tfrac{c}{2a} = 0}}, then substituting the new coefficients into the standard quadratic formula. Because this variant allows re-use of the intermediately calculated quantity {{tmath|\tfrac{b}{2a} }}, it can slightly reduce the arithmetic involved.

=== Square root in the denominator <span class="anchor" id="Citardauq"></span> ===
A lesser known quadratic formula, first mentioned by [[Giulio Carlo de' Toschi di Fagnano|Giulio Fagnano]],<ref>Specifically, Fagnano began with the equation {{tmath|1= xx + bb = ax}} and found the solutions to be <math display=block>x = \frac{2 bb}{a \mp a \sqrt{1 - \dfrac{4bb}{aa}}}.</math> (In the 18th century, the square {{tmath|\textstyle x^2}} was conventionally written as {{nobr|{{tmath|xx}}.)}} {{pb}}
{{citation |last=Fagnano |first=Giulio Carlo |author-link=Giulio Carlo de' Toschi di Fagnano |title=Produzioni matematiche del conte Giulio Carlo di Fagnano, Marchese de' Toschi, e DiSant' Ononio |volume=1 |place=Pesaro |publisher=Gavelliana |year=1750 |chapter=Applicazione dell' algoritmo nuovo Alla resoluzione analitica dell' equazioni del secondo, del terzo, e del quarto grado |trans-chapter=Application of a new algorithm to the analytical resolution of equations of the second, third, and fourth degree |language=it |at=Appendice seconda, eq. 6, {{pgs|467}} |doi=10.3931/e-rara-8663 |doi-access=free |chapter-url=https://era-prod11.ethz.ch/zut/content/zoom/2406151 }}</ref> describes the same roots via an equation with the square root in the denominator (assuming {{tmath|c \neq 0}}):

<math display=block>
x= \frac{2c}{-b \mp \sqrt {b^2 - 4ac}}.
</math>

Here the [[minus–plus symbol]] "{{tmath|\mp}}" indicates that the two roots of the quadratic equation, in the same order as the standard quadratic formula, are

<math display=block>
x_1 = \frac{2c}{-b - \sqrt {b^2 - 4ac}}, \qquad
x_2 = \frac{2c}{-b + \sqrt {b^2 - 4ac}}.
</math>

This variant has been jokingly called the "citardauq" formula ("quadratic" spelled backwards).<ref>{{citation |last=Goff |first=Gerald K. |year=1976 |title=The Citardauq Formula |journal=The Mathematics Teacher |volume=69 |number=7 |pages=550–551 |doi=10.5951/MT.69.7.0550 |jstor=27960584 }}</ref>

When {{tmath|-b}} has the opposite sign as either {{tmath|\textstyle +\sqrt {b^2 - 4ac} }} or {{tmath|\textstyle -\sqrt {b^2 - 4ac} }}, subtraction can cause [[catastrophic cancellation]], resulting in poor accuracy in numerical calculations; choosing between the version of the quadratic formula with the square root in the numerator or denominator depending on the sign of {{tmath|b}} can avoid this problem. See {{slink|#Numerical calculation}} below.

This version of the quadratic formula is used in [[Muller's method]] for finding the roots of general functions. It can be derived from the standard formula from the identity {{tmath|1= x_1 x_2 = c/a}}, one of [[Vieta's formulas]]. Alternately, it can be derived by dividing each side of the equation {{tmath|1=\textstyle ax^2 + bx + c = 0}} by {{tmath|\textstyle x^2}} to get {{tmath|1=\textstyle cx^{-2} + bx^{-1} + a = 0}}, applying the standard formula to find the two roots {{tmath|\textstyle x^{-1}\! }}, and then taking the reciprocal to find the roots {{tmath|x}} of the original equation.