A transformation that reduces coefficient count to one greatly simplifies the task of finding roots.

**Edwin P. Russo, P.E.**Professor Emeritus

National Center for Advanced Mfg.

**Carsie A. Hall **

Associate Professor

**Duane J. Jardine**Adjunct Professor

Mechanical Engineering Dept.

University of New Orleans

New Orleans, La.

**William W. St. Cyr, P.E.**NASA Stennis Space Center

Stennis, Miss.

The general cubic equation given by:

(1)has roots that are solvable by classical techniques involving the computation of inverse cosines, cosines of multiple angles, and so forth. But equation 1 may be transformed into a simpler equation with only one coefficient. Such a transformation resembles those given elsewhere, though the other transformations only reduce the number of coefficients from three to two, rather than to one.

Jahnke and Emde in 1945 derived a formulation giving a single coefficient with a short table of roots. But it needed a highly complicated graphical procedure to find the roots. The transformation presented here as well reduces the coefficient count to one but gives a convenient means by which to tabularize the roots of the cubic equation, eliminating various approximate or tedious methods of finding the roots.

Equation 1 may be transformed into:

(2)

where

(3)

or

(4)

and

(5)

The above transformation first defines a new variable, *X*, by the relation:

(6)

where *D *and *K *are arbitrary constants to be determined later. Substituting equation 6 into equation 1 gives:

(7)

The *X *^{2 }term is eliminated by requiring that *K = a/3. D *is now defined by requiring that the coefficients of *X *^{3 }and *X *be the same. Therefore:

(8)

or, after substituting *K*:

(9)

Dividing equation 7 by *D*^{3 }and substituting for *K *gives an equation with only one coefficient, that is, equation 2. This technique also works to reduce the number of coefficients for higher-order equations (quartic, for example).

An abbreviated Table of roots contains real values of *P *in rather large increments. Interested readers may wish to expand the Table to include finer increments of *P*. Note that the real part of the complex roots, *X2 *and *X3*, is simply *X1/2, *and that *X2 *and *X3 *are complex conjugates. Examination of equation 5 shows that if *a*, *b*, and *c *are real, then the coefficient *P *can only be real (positive or negative) or imaginary if *b < a *^{2 }*/*3.

When *P *is imaginary (*b < a *^{2 }*/*3) the cubic equation 2 may be rewritten in a more convenient form, namely:

where *P *has been replaced by *iQ *and *X *replaced by *iY *(*Q *is real and i = √–1). That is:

(11)

or

(12)

and

(13)

Table 2 lists the roots of equation 10 for various values of the coefficient *Q*. Only when *Q *is in the interval, –4/27 < Q < √4/27, will the roots of equation 10 all be real. These roots are listed in Table 3.

After selecting the roots of equation 2 or 10 from the appropriate Table, it is a simple matter to obtain the roots for the general cubic equation 1 from equation 4 or 12.

Note when *P *is large:

And when *Q *is large:

The Tables may be used to solve principal stresses, pump curves, eigenvalues, hydraulic jumps, control systems, spillway flow, moving wave/bores, setting initial conditions for Newton iterations, and so on.