### Investigating Cardano’s Irreducible Case

#### Abstract

Solving cubic equations is a historically rich problem in mathematics. Unlike with quadratic equations, cubic equations do not have a “cubic formula.” However, over the years many techniques have been presented that often find the solutions of cubic equations. This research investigates one of these techniques known as Cardano’s Method. This method provides an algebraic technique for solving the general cubic equation. Since its inception, this technique has suffered a significant drawback. In some instances, the application of Cardano’s Method results in what Cardano termed the “irreducible case.” The irreducible case occurs when a complex number is needed to complete the process. This research investigates the relationship among the coefficients of the general cubic equation and the irreducible case, and it has determined that these relationships fall into one of three categories: always reducible, always irreducible, or conditionally irreducible. This research has discovered which relationships fall into each of the aforementioned categories. One possible result of this research is the formulation of a general algorithm to easily determine whether a given cubic equation will produce Cardano’s irreducible case.

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