An alternative way to solve the general cubic, from Dan Kalman 
and James White.      

Here follows an alternative way to derive a solution to the cubic equation 
rediscovered by Kalman and White. A closely related approach was presented by Oglesby in 1923.

Any arbitrary cubic equation can be reduced to one of the form

x3 + px + q = 0                                                                                     (1)

by a linear change of variable. So in what follows, we will only consider this kind of cubic equation.
The derivation that we will present depends on the following identity.

(wa + b + c)(a + wb + c)(a + b + wc) = (a3 +b3 +c3)w - 3abcw2          (2)

 Here, a; b; and c are arbitrary complex constants and                   w   = 1+iÖ3     w is a cube root of 1, 
                                            2 
and so satisfies a number of identities:  w3 = 1,   w2+w+1 = 0,   w + 1 = -w2

To verify Eq. (2), simply multiply out the left side, collect like monomials in a; b; and c,and apply the 
identities for w listed above. Symmetry simplies the process considerably.

Collecting together all terms involving a2b results in a coefficient off w2 + w+1 = 0. By symmetry, the 
terms involving a2c, b2c, etc., also vanish. In the expansion of the lefthand side of the identity, that leaves 
only terms involving a3; b3; c3; and abc; and by considering the coefficients of these terms, Eq. (2) is 
easily established.To render the identity more recognizable, replace a with x; which is to be thought of 
as the variable of the cubic. 
That produces (wx + b + c)(x + wb + c)(x + b + wc) = (x3 +b3 +c3)w - 3abcw2       Factoring out w on 
the right and rerranging the remaining terms then leads to:   

(wx + b + c)(x + wb + c)(x + b + wc) = w(x3  - 3xbcw + b3 + c3)        (3) 

Now we can recognize that the right side is essentially the same as what appears in Eq. (1), provided 
that the following relations hold:

             -3bcw = p                                                                                (4)
            b3 + c3 = q                                                                                (5)

Given values of p and q; we need only determine a, b and c satisfying these relations, whereupon Eq. (3) 
provides a factorization to linear factors. Fortunately, we can solve (2) for b and c in a straightforward 
way. Indeed, if the original system of equations is rewritten in the following form:     
                        b3c3       = - p3/27 
                        b3 + c3 = q

it is immediately apparent that b3 and c3 are the roots of the quadratic equation x2 + qx + p3/27 = 0;  and 
are given by[q ±  Ö( q2 + 4p3/27)]1/3
                                 2

Note here that when p and q are real, we obtain real values for b3 and c3 just when q2 + 4p3=27 ¸ 0. 
That leads to

 b = [q +  Ö( q2 + 4p3/27)]1/3
                     2

c = [q  -  Ö( q2 + 4p3/27)]1/3 
                     2

When the equations for b3 and c3 produce complex (that is, non-real) values, we have to be a bit 
more careful. There are three complex cuberoots among which to choose b and c, and not all 
combinations satisfy the original equations for b and c. While it is clear that Eq. (5) will be satisfied 
in any case, Eq. (4) requires that consistent values of b and c be selected. For this situation, we can 
choose any of the three complex cuberoots for b, and then define c as  - p/(3wb). To complete the 
solution of the cubic, we note that the solutions to Eq. (1) must also be  
roots of  (wx + b + c)(x + wb + c)(x + b + wc) = 0

By inspection, the solutions are:

x = -(b + c)=w;      x = -(wb + c)     x = -(b + wc)

This result is closely related to, but slightly different from the standard solution to the cubic that has 
been handed down with little if any modification since it was published by Cardano in 1545. Although 
it was originally derived by a different method, Cardano's solution can be formulated in terms of the 
following identity: 

(a + b + c)(a + wb + w2c)(a + w2b + wc) = a3 +b3 +c3 -3abc                                    (6)

 This identity has appeared in earlier papers ([6, 9]) on the solution of cubic equations. It is very similar 
to Eq. (2), from which it can be derived by replacing a with a=w.From Eq. (6), virtually the same steps 
presented above lead to the traditional form of Cardano's solution to the cubic. The symmetry of Eq. (2) 
may make its  verification somewhat simpler than the verification of Eq. (6). Otherwise, either identity 
provides a simple approach to solving the cubic.A streamlined version of the Cardano solution is 
particularly simple and memorable.Following the presentation in [10], begin with the cubic in the 
form x3 = px + q and replace x with b + c. That leads directly to the equation

3bcx + b3 + c3 = px + q

Now match the coefficients of x : make 3bc = p and b3 + c3 = q: These conditions are virtually 
identical to those used earlier, and allow us to find b and c in terms of p and q:

From this point on, the derivation is essentially the same as what was presented before. There is a large 
literature on solving the cubic. References [3], [4], [9], and [11] are representative samples. Besides 
the previously cited presentation in [10], there is a recently published version in [5]. A translation of 
Cardano's published solution appears in [2]. Kleiner [7] provides an interesting discussion of the role 
of the solution of the cubic in the development of complex numbers. There is also an interesting history 
associated with Cardano's publication and his dispute with Tartaglia [1, 8].

 

Link to  the source: 

http://www.american.edu/academic.depts/cas/mathstat/People/kalman/pdffiles/Newcubic.pdf

References

[1] Carl Boyer. A History of Mathematics, 2nd edition, revised by Uta C. Merzbach,Wiley, 
New York, 1991, pp 282 { 286.

[2] Ronald Calinger, ed. Classics of Mathematics, Moore Publishing, Oak Park, Illinois,1982, 
pp 235 { 237.

[3] H. B. Curtis. A derivation of Cardan's formula. American Mathematical Monthly, 51(1), 
January, 1944, p. 35.

[4] Orrin Frink Jr. A method for solving the cubic. American Mathematical Monthly,32(3), March, 
1925, p. 134.

[5] Jan Gullberg. Mathematics from the birth of numbers, W. W. Norton, New York,1997, p. 318.

[6] Morton J. Hellman. A unifying technique for the solution of the quadratic, cubic, and quartic. 
American Mathematical Monthly, 65(4), April, 1959, pp. 274 { 276.

[7] Isreal Kleiner. Thinking the Unthinkable: The story of the complex numbers (with amoral). 
Mathematics Teacher, 81, October, 1988, pp. 583 { 592. Reprinted in Frank

J. Swetz, ed., From Øve Øngers to inØnity, Open Court, Chicago and La Salle, Ill,1994, 
pp. 711 { 720.

[8] Victor J. Katz. A History of Mathematics - an Introduction, Harper Collins, NewYork, 
1993, pp 330 { 334.

[9] E. J. Oglesby. Note on the algebraic solution of the cubic. American Mathematical Montly, 
30(6), Sept - Oct, 1923, pp. 321 { 323.

[10] John Stillwell. Mathematics and its History, Springer-Verlag, New York, 1989, pp 55.

[11] Robert Y. Suen. Roots of cubics via determinants. College Mathematics Journal, 25(2), 
March, 1994, pp. 115 { 117.

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