Explicit expressions for real roots of a quartic equation

TL;DR

This paper provides explicit, optimized formulas for the real roots of a quartic equation, simplifying the general solution specifically for cases with four real roots.

Contribution

It introduces a shorter, more efficient set of explicit expressions for real roots of quartic equations, improving upon the general formulas.

Findings

Derived explicit formulas for all four real roots of a quartic.

Presented a shorter, optimized expression compared to the general quartic solution.

Enhanced computational efficiency for solving quartics with four real roots.

Abstract

This short article presents explicit expressions for roots of a quartic equation that has all four real roots. Although a general expression for quartic roots is available on Wikipedia, an optimized and slightly shorter expression for only real roots is presented here.

Equations

$$x^4+a{x}^3+b{x}^2+cx+d=0.$$

$$y^4+p{y}^2+qy+r=0,$$

$$p=b-6{\left(\frac{a}{4}\right)}^2,$$

$$q=c-2b\left(\frac{a}{4}\right)+8{\left(\frac{a}{4}\right)}^3,$$

$$r=d-c\left(\frac{a}{4}\right)+b{\left(\frac{a}{4}\right)}^2-3{\left(\frac{a}{4}\right)}^4.$$

$${\left(y^2+\frac{p}{2}\right)}^2=-qy+\frac{p^2}{4}-r.$$

$${\left(y^2+\frac{p}{2}+z\right)}^2=2z{y}^2-qy+z^2+zp+\frac{p^2}{4}-r.$$

$$8z^3+8p{z}^2+(2p^2-8r)z-q^2=0,$$

$${\left(y^2+\frac{p}{2}+z_0\right)}^2={\left(y\sqrt{2z_0}-\frac{q}{2\sqrt{2z_0}}\right)}^2.$$

$$\left(y^2+\sqrt{2z_0}y+\frac{p}{2}+z_0-\frac{q}{2\sqrt{2z_0}}\right)\left(y^2-\sqrt{2z_0}y+\frac{p}{2}+z_0+\frac{q}{2\sqrt{2z_0}}\right)=0,$$

$$x_{1,2}=-\frac{a}{4}-\frac{1}{2}\sqrt{2z_0}\pm \frac{1}{2}\sqrt{-\left(2p+2z_0-\frac{2q}{\sqrt{2z_0}}\right)},$$

$$x_{3,4}=-\frac{a}{4}+\frac{1}{2}\sqrt{2z_0}\pm \frac{1}{2}\sqrt{-\left(2p+2z_0+\frac{2q}{\sqrt{2z_0}}\right)}.$$

$$x^3+\alpha x^2+\beta x+\gamma =0,$$

$$x_0=s_1+s_2-\frac{\alpha }{3},$$

$$s_1=\sqrt[3]{R+\sqrt{R^2+Q^3}},$$

$$s_2=\sqrt[3]{R-\sqrt{R^2+Q^3}},$$

$$Q=\frac{3\beta -{\alpha }^2}{9},$$

$$R=\frac{9\alpha \beta -27\gamma -2{\alpha }^3}{54}.$$

$$2z_0=\frac{1}{3}\left(S+\frac{{\Delta }_0}{S}-2p\right),$$

$$S=6s=\sqrt[3]{\frac{{\Delta }_1+\text{sign}({\Delta }_1)\sqrt{{\Delta }_1^2-4{\Delta }_0^3}}{2}},$$

$${\Delta }_0=-36Q=b^2+12d-3ac,$$

$${\Delta }_1=432R=27a^{2}d-9abc+2b^3-72bd+27c^2.$$

$$z_0=\frac{1}{3}\left(\sqrt{{\Delta }_0}\cos \frac{\varphi }{3}-p\right),$$

$$\varphi =\arccos \left(\frac{{\Delta }_1}{2\sqrt{{\Delta }_0^3}}\right).$$

$${\lambda }_{1,4}=-\frac{a}{4}\mp \frac{\sqrt{Z}+\sqrt{-A-Z\pm \frac{B}{\sqrt{Z}}}}{2},$$

$${\lambda }_{2,3}=-\frac{a}{4}\mp \frac{\sqrt{Z}-\sqrt{-A-Z\pm \frac{B}{\sqrt{Z}}}}{2}.$$

$$Z=2z_0=\frac{1}{3}\left(2\sqrt{{\Delta }_0}\cos \frac{\varphi }{3}-A\right),$$

$$\varphi =\arccos \left(\frac{{\Delta }_1}{2{\Delta }_0\sqrt{{\Delta }_0}}\right),$$

$$A=2p=2b-\frac{3a^2}{4},$$

$$B=2q=2c-ab+\frac{a^3}{4}.$$

Full text

Explicit expressions for real roots of a quartic equation

Nino Krvavica

(University of Rijeka, Faculty of Civil Engineering

nino.krvavica@uniri.hr

)

This short article presents explicit expressions for roots of a quartic equation that has all four real roots. Although a general expression for quartic roots is available on Wikipedia [4], an optimized and slightly shorter expression for only real roots is presented here. A derivation of closed-form solutions for real roots of a quartic presented in the first part of this paper is taken from [2]. It is repeated here for completeness and clarity.

Let us consider a general normalized 4th order polynomial equation (quartic)

[TABLE]

To find the analytical solution to roots of Eq. (1), first the cubic term $x^{3}$ is eliminated and the general polynomial is converted into a so-called depressed quartic by a change of variables. Following Ferrari’s method [1], a substitution $x=y-a/4$ is introduced, which gives a depressed polynomial

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

The depressed polynomial can be rewritten as

[TABLE]

Next, expression $2zy^{2}+zp+z^{2}$ is added to both sides of Eq. (6), which after some regrouping gives

[TABLE]

When $z$ is chosen to be any non-zero root $z_{0}$ of the so-called resolvent cubic equation

[TABLE]

the right-hand side of Eq. (7) can be written as a perfect square; therefore, Eq. (7) becomes

[TABLE]

And finally, Eq. (9) can be written as a factorized quadratic equation

[TABLE]

which is easily solved by a quadratic formula.

Therefore, the solutions to the roots of the general quartic Eq. (1) are given by

[TABLE]

[TABLE]

For a general normalized 3rd order polynomial equation (cubic)

[TABLE]

a real solution is given by Cardano’s formula [1]

[TABLE]

with

[TABLE]

[TABLE]

where

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[TABLE]

To eliminate redundant divisions and optimize computation of Eq. (11) and (12), the root of the resolvent cubic equation is expressed via

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Note that $\Delta_{1}^{2}-4\Delta_{0}^{3}$, which is a much simpler expression for the discriminant of the resolvent cubic equation $\mathcal{D}_{cubic}$ and especially the discriminant of the quartic equation $\mathcal{D}_{quartic}$. Therefore, if $\Delta_{1}^{2}-4\Delta_{0}^{3}<0$, three resolvent cubic roots are all real and the quartic roots are either all complex or all real. In this case, Eq. (19) can be solved trigonometrically [3], which is computationally faster than computing the cube root required in Eq. (20):

[TABLE]

where

[TABLE]

To summarize, a closed-form real solutions to the quartic equation can be simplified as follows:

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

with

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[TABLE]

Finally, by including Eqs. (27)-(30) and Eqs (21) and (22) into Eqs. (25) and (26), the fully explicit form of quartic roots is obtained:

[TABLE]

[TABLE]