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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 equation

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).

Ferrari was the first to develop an algebraic technique for solving the general quartic, which was stolen and published in Cardano's Ars Magna in 1545 (Boyer and Merzbach 1991, p. 283).

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.

Cite this as:

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

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