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Quartic Formula

The quartic formula is a name sometimes given to one of the related explicit formulas for the four roots z_1, ..., z_4 of an arbitrary quartic equation with real coefficients

z^4+a_3z^3+a_2z^2+a_1z+a_0=0.
(1)

Let y_1 be a real root of the resolvent cubic

y^3-a_2y^2+(a_1a_3-4a_0)y+(4a_2a_0-a_1^2-a_3^2a_0)=0.
(2)

Then one of its most common forms of the quartic formula is given by

z_1=-1/4a_3+1/2R+1/2D
(3)
z_2=-1/4a_3+1/2R-1/2D
(4)
z_3=-1/4a_3-1/2R+1/2E
(5)
z_4=-1/4a_3-1/2R-1/2E,
(6)

where

R=sqrt(1/4a_3^2-a_2+y_1)
(7)
D={sqrt(3/4a_3^2-R^2-2a_2+1/4(4a_3a_2-8a_1-a_3^3)R^(-1)) for R!=0; sqrt(3/4a_3^2-2a_2+2sqrt(y_1^2-4a_0)) for R=0
(8)
E={sqrt(3/4a_3^2-R^2-2a_2-1/4(4a_3a_2-8a_1-a_3^3)R^(-1)) for R!=0; sqrt(3/4a_3^2-2a_2-2sqrt(y_1^2-4a_0)) for R=0
(9)

(Abramowitz and Stegun 1972, p. 17; Beyer 1987a, p. 12).

Cardano (1545, Ch. 39, fol. 72v) published an algebraic method for solving quartic equations and explicitly credited Ferrari.

See also

Quadratic Equation, Quadratic Formula, Quartic Equation

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Solutions of Quartic Equations." §3.8.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 17-18, 1972.Berger, M. §16.4.1-16.4.11.1 in Geometry I. New York: Springer-Verlag, 1987.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 12, 1987a.Beyer, W. H. Handbook of Mathematical Sciences, 6th ed. Boca Raton, FL: CRC Press, 1987b.Cardano, G. Artis magnae, sive de regulis algebraicis, liber unus. Nuremberg, Germany: Petreius, 1545. https://doi.org/10.3931/e-rara-9159.

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Quartic Formula

Cite this as:

Weisstein, Eric W. "Quartic Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuarticFormula.html

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