Introduction and main result
By 1494, cubic equations were in general unsolvable algebraically. The first abstract solutions can be found in the work of the Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, Girolamo Cardano, and Rafael Bombelli (cf. [1], Chapter 12: Algebra in the Renaissance, p.383ff). Since Cardano was the first to publish the corresponding results in 1545, the formulas for the solution found at that time are named after him.
Another simple approach to an algebraic solution of cubic equations can be found in [2], p. 134 ff, analyzing the divisors of the integer constant term in the equation motivated by Vieta’s theorem. A completely different way to solve cubic equations “algebraically” can be found in [3], p. 235, which, however, turns out to be only a very good numerical approximation of the solution.
To be more precise, we consider the equation
for the case
with rational a, b This is the case where exactly one root is real and the other ones are complex. According to Cardano’s formula (cf. [3], p. 233), we can write the real solution x as
where
(1)
However, even if the solution x is rational, w1 and w1 are typically irrational. We want to show in this note that in this case w1 and w2 can surprisingly be expressed as terms solely depending on square roots of rational numbers, i.e.
where
(2)
with
This problem has also been addressed in [4] p. 163ff. On the contrary, our paper presents a complete explicit solution for all relevant cases.
Example 1. Choose
and
Then the resulting equation is
with the only real solution
According to Cardano’s formula, we have
and
(3)
with the numerical approximation
With
and
we obtain
and
(4)
with the exact solution
Proof of alternate solution form
We write for short
Then
and
hence
or
(6)
with the solution (note
We put
and equate the above equations to obtain
and
This means that we have to solve
Rewriting this as
we get by subtraction
Example 2. Consider the equation
with the only real solution
This can in general not be expressed solely with square roots of rational numbers.
Note that the statements above are not only restricted to Cardano’s case but apply also to the casus irreducibilis where a3 + b2 < 0 with rational a, b In this case we have only real solutions in spite of the fact that the roots in Cardano’s formula are complex. We only have to keep in mind that w1 and w2 have in general three different representations as complex numbers. If w1 and w2 are any given complex values then the other two are obtained by multiplications with the two non-trivial unit roots
and
Example 3. Consider the equation
with
and
Note that
is a rational solution to the cubic equation. Cardano’s formula gives
and
(13)
with the three corresponding numerical complex representations
with the three numerical approximations
. Our alternative approach gives
and s = −12 with
and
with the exact solution
Conclusion
It is to be noted that although in case of a rational solution to the cubic equation relation (2) above does not provide an algorithmic approach to find the root of the equation.