Quartic Equation

A quartic equation is a fourth-order polynomial equation of the form

(1)

While some authors (Beyer 1987b, p. 34) use the term "biquadratic equation" as a synonym for quartic equation, others (Hazewinkel 1988, Gellert et al. 1989) reserve the term for a quartic equation having no cubic or linear term, i.e., a quadratic equation in .

Cardano (1545, Ch. 39, fol. 72v) published an algebraic method for solving quartic equations and explicitly credited Ferrari. The Wolfram Language can solve quartic equations exactly using the built-in command Solve[a4 x^4 + a3 x^3 + a2 x^2 + a1 x + a0 == 0, x]. The solution can also be expressed in terms of Wolfram Language algebraic root objects by first issuing SetOptions[Roots, Quartics -> False].

The roots of this equation satisfy Vieta's formulas:

	(2)
	(3)
	(4)
	(5)

The substitution gives the depressed form, which has no cubic term,

(6)

whose roots are denoted by , ..., . The properties of the symmetric polynomials appearing in Vieta's formulas then give

			(7)
			(8)
			(9)
			(10)

The depressed roots also satisfy the pairwise relations

(11)

(12)

(13)

as well as their cyclic permutations.

A worked example in Cardano (1545, Ch. 39, fol. 73v), also explicitly credited to Ferrari, is

(14)

(Smith 1994, p. 207).

The term can be eliminated from the general quartic (1) by making a substitution of the form

(15)

x^4+(a_3-4lambda)x^3+(a_2-3a_3lambda+6lambda^2)x^2
+(a_1-2a_2lambda+3a_3lambda^2-4lambda^3)x+(a_0-a_1lambda+a_2lambda^2-a_3lambda^3+lambda^4)=0.

(16)

Letting so

(17)

then gives the standard form

(18)

where

			(19)
			(20)
			(21)

The quartic can be solved by writing it in a general form that would allow it to be algebraically factorable and then finding the condition to put it in this form. The equation that must be solved to make it factorable is called the resolvent cubic. To do this, note that the quartic will be factorable if it can be written as a difference of squares,

(22)

It turns out that a polynomial factorization of this form can be obtained by adding and subtracting (where is for now an arbitrary quantity, but which will be specified shortly) to equation (6) to obtain

(23)

This equation can be rewritten

(24)

(Birkhoff and Mac Lane 1966). Note that the first term is immediately a perfect square with

(25)

For , the second term will be a perfect square if is chosen so that the square can be completed in

(26)

This means we want

(27)

which requires that

(28)

(29)

This is the resolvent cubic. If , it forces , and the depressed quartic is a biquadratic equation.

For a solution of equation (29), substituting the resolvent condition into equation (26) gives

(30)

with

(31)

therefore is linear in and is quadratic in , so each term and is quadratic and can be solved using the quadratic formula. These give the four roots of the depressed quartic. The roots of the original quartic follow from .

Explicitly, plugging , , and back into (29) gives

u^3+(3/8a_3^2-a_2)u^2+(3/(64)a_3^4-1/4a_2a_3^2+a_1a_3-4a_0)u+(1/(512)a_3^6-1/(64)a_2a_3^4+1/8a_1a_3^3-3/2a_0a_3^2+4a_0a_2-a_1^2)=0.

(32)

This can be simplified by making the substitution

(33)

which gives the resolvent cubic

(34)

Let be a root of (34). With defined below, the four roots of the original quartic are obtained by solving the two quadratic equations

$0={z^2+(1/2a_3+/-R)z+1/2[y_1+/-(a_3y_1/2-a_1)/R] for R!=0; z^2+1/2a_3z+1/2(y_1+/-sqrt(y_1^2-4a_0)) for R=0,$

(35)

with the signs correlated within each row. For , substituting , , and into and gives the first row. The resolvent cubic implies , so replacing by an independently chosen square root can give the wrong sign. If , the same identity forces and gives the second row by a difference of squares. Applying the quadratic formula to these factors gives

			(36)
			(37)
			(38)
			(39)

where

			(40)
		${sqrt(3/4a_3^2-R^2-2a_2+1/4(4a_3a_2-8a_1-a_3^3)R^(-1)) for R!=0; sqrt(3/4a_3^2-2a_2+2sqrt(y_1^2-4a_0)) for R=0$	(41)
		${sqrt(3/4a_3^2-R^2-2a_2-1/4(4a_3a_2-8a_1-a_3^3)R^(-1)) for R!=0; sqrt(3/4a_3^2-2a_2-2sqrt(y_1^2-4a_0)) for R=0$	(42)

(Abramowitz and Stegun 1972, p. 17; Beyer 1987a, p. 12). This result is sometimes known as the quartic formula.

Another approach uses the roots , ..., of the depressed quartic (6) to define

			(43)
			(44)
			(45)

where the second forms follow from

(46)

and defining

			(47)
			(48)

This equation can be written in terms of the coefficients , , and of the depressed quartic as

(49)

The roots of this cubic equation then give , , and , and the equations (43) to (45) can be solved for the four roots of the depressed quartic (Faucette 1996). For an explicit reconstruction, choose square roots , , and with . The roots are then

			(50)
			(51)
			(52)
			(53)

For , either sign may be chosen for each of and , after which set . The identity ensures that this choice satisfies . For , at least one of the square roots is zero, so their product is already zero and no division is needed. Finally, the roots of the original quartic are (NIST DLMF, §1.11(iii)).

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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.Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 107-108, 1996.Borwein, P. and Erdélyi, T. "Quartic Equations." §1.1.E.1e in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, p. 4, 1995.Boyer, C. B. and Merzbach, U. C. A History of Mathematics, 2nd ed. New York: Wiley, pp. 286-287, 1991.Cardano, G. Artis magnae, sive de regulis algebraicis, liber unus. Nuremberg, Germany: Petreius, 1545. https://doi.org/10.3931/e-rara-9159.Ehrlich, G. §4.16 in Fundamental Concepts of Abstract Algebra. Boston, MA: PWS-Kent, 1991.Faucette, W. M. "A Geometric Interpretation of the Solution of the General Quartic Polynomial." Amer. Math. Monthly 103, 51-57, 1996.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989.Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, 1988.MathPages. "Reducing Quartics to Cubics." https://www.mathpages.com/home/kmath296.htm.National Institute of Standards and Technology. "Polynomials of Degrees Two, Three, and Four." §1.11(iii) in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/1.11.iii.Smith, D. E. A Source Book in Mathematics. New York: Dover, 1994.van der Waerden, B. L. §64 in Algebra, Vol. 1. New York: Springer-Verlag, 1993.

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Quartic Equation

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Quartic Equation

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