Quadratic formula
Square root in the denominator

A lesser known quadratic formula, first mentioned by Giulio Fagnano, describes the same roots via an equation with the square root in the denominator (assuming c ≠ 0 {\displaystyle c\neq 0}):

x = 2 c − b ∓ b 2 − 4 a c. {\displaystyle x={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac}}}}.}

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

x 1 = 2 c − b − b 2 − 4 a c, x 2 = 2 c − b + b 2 − 4 a c. {\displaystyle x_{1}={\frac {2c}{-b-{\sqrt {b^{2}-4ac}}}},\qquad x_{2}={\frac {2c}{-b+{\sqrt {b^{2}-4ac}}}}.}

This variant has been jokingly called the "citardauq" formula ("quadratic" spelled backwards).

When − b {\displaystyle -b} has the opposite sign as either + b 2 − 4 a c {\displaystyle \textstyle +{\sqrt {b^{2}-4ac}}} or − b 2 − 4 a c {\displaystyle \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 b {\displaystyle b} can avoid this problem. See § 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 x 1 x 2 = c / a {\displaystyle x_{1}x_{2}=c/a}, one of Vieta's formulas. Alternately, it can be derived by dividing the a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} by x 2 {\displaystyle \textstyle x^{2}} to get c x − 2 + b x − 1 + a = 0 {\displaystyle \textstyle cx^{-2}+bx^{-1}+a=0}, applying the standard formula to find the two roots x − 1 {\displaystyle \textstyle x^{-1}\!}, and then taking the reciprocal to find the roots x {\displaystyle x} of the original equation.
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