Cubic Formula

The cubic formula is the closed-form solution for a cubic equation, i.e., the roots of a cubic polynomial. A general cubic equation is of the form

(the coefficients , , and may be complex, and the coefficient of may be taken as 1 without loss of generality by dividing the entire equation through by , where ). The Wolfram Language can solve cubic equations exactly using the built-in command Solve[a3 x^3 + a2 x^2 + a1 x + a0 == 0, x]. The solution can also be expressed in terms of the Wolfram Language algebraic root objects by first issuing SetOptions[Roots, Cubics -> False].

The solution to the cubic (as well as the quartic) was published by Gerolamo Cardano (1501-1576) in his treatise Ars Magna. However, Cardano was not the original discoverer of either of these results. The hint for the cubic had been provided by Niccolò Tartaglia, while the quartic had been solved by Ludovico Ferrari. However, Tartaglia himself had probably caught wind of the solution from another source. The solution was apparently first arrived at by a little-remembered professor of mathematics at the University of Bologna by the name of Scipione del Ferro (ca. 1465-1526). While del Ferro did not publish his solution, he disclosed it to his student Antonio Maria Fior (Boyer and Merzbach 1991, p. 283). This is apparently where Tartaglia learned of the solution around 1541.

To solve the general cubic (1), it is reasonable to begin by attempting to eliminate the term by making a substitution of the form

Then

The is eliminated by letting , so

Then

so equation (1) becomes

Defining

then allows the shifted cubic to be written in the standard form

The simplest way to proceed is to make Vieta's substitution

with , which reduces the cubic to the equation

which is easily turned into a quadratic equation in by multiplying through by to obtain

(Birkhoff and Mac Lane 1996, p. 106). The result from the quadratic formula is

where and . Unless , choose one nonzero value of from the quadratic formula. Substituting its three cube roots into (12) gives the three roots of the cubic equation, counted with multiplicity. When , the cubic equation has the triple root , and (12) is undefined at .

Equation (11) may also be explicitly factored by attempting to pull out a term of the form from the cubic equation, leaving behind a quadratic equation which can then be factored using the quadratic formula. This process gives the same solutions as Vieta's substitution. First, define the intermediate variables

(which are identical to and up to a constant factor). The general cubic equation (11) then becomes

Let and be, for the moment, arbitrary constants. An identity satisfied by perfect cubic polynomial equations is that

The general cubic would therefore be directly factorable if it did not have an term (i.e., if ). However, since in general , add a multiple of --say --to both sides of (19) to give the slightly messy identity

which, after regrouping terms, is

We would now like to match the coefficients and with those of equation (18), so we must have

Plugging the former into the latter then gives

Therefore, if we can find a value of satisfying the above identity, we have factored a linear term from the cubic, thus reducing it to a quadratic equation. Writing gives . Choosing and therefore gives the symmetrical expression

The cube roots in this expression must be paired so that . All fractional powers in the following expansions denote powers or products of these paired cube roots. Taking the second and third powers of gives

Plugging the expression for into the left side of (24) gives

so we have indeed found the factor of (18), and we need now only factor the quadratic part. Plugging into the quadratic part of (21) and solving the resulting

then gives the solutions

These can be simplified by defining

so that the solutions to the quadratic part can be written

Define

Here is an auxiliary discriminant. The polynomial discriminant of the cubic is related to by (Birkhoff and Mac Lane 1996). The cube roots must be chosen so that . For , this can be enforced by setting rather than choosing the two principal roots independently. This gives very simple expressions for and , namely

Therefore, at last, the roots of the original equation in are then given by

with the coefficient of in the original equation, and and as defined above. These three equations giving the three roots of the cubic equation are sometimes known as Cardano's formula. Note that if the equation is in the standard form of Vieta

in the variable , then , , and , and the intermediate variables have the simple form (cf. Beyer 1987)

The solutions satisfy Vieta's formulas

In standard form (11), , , and , so eliminating gives

for , and eliminating gives

for . In addition, the properties of the symmetric polynomials appearing in Vieta's formulas give

The discriminant classification and the trigonometric and hyperbolic formulas below assume that the coefficients , , and are real. The equation for in Cardano's formula does not have an appearing in it explicitly while and do, but this does not say anything about the number of real and complex roots (since and are themselves, in general, complex). However, determining which roots are real and which are complex can be accomplished by noting that if the polynomial discriminant , one root is real and two are complex conjugates; if , all roots are real and at least two are equal; and if , all roots are real and unequal. Equivalently, these three cases have , , and , respectively. If , define

Then the real solutions are of the form

If in the standard form (11), its real solution is . For , the trigonometric approach can be generalized using the identity

(Dickson 1914) and setting

(Birkhoff and Mac Lane 1996, pp. 90-91), then

If , then use

to obtain

If and , use

and if and , use

to obtain

The solutions to the original equation are then

An alternate approach to solving the cubic equation is to use Lagrange resolvents (Faucette 1996). Here denotes the Lagrange resolvent , where is a cube root of unity. Let , define

where are the roots of

and consider the equation

where and are complex numbers. The roots are then

for , 1, 2. Multiplying through gives

which matches the depressed cubic equation (11), provided that

Some curious identities involving the roots of a cubic equation due to Ramanujan are given by Berndt (1994).