Now, Cardan's formula has the drawback that it may bring such square roots into play in intermediate steps of computation, even when those numbers do not appear in the problem or its answer. None of this material was discovered by me. With higher-degree polynomials, factoring can be even more difficult.
There are other functions that would also work, and some of them are more interesting to mathematicians for various reasons, but I like this one because it can be described in fairly elementary terms.
In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. To do this, though, we need an efficient method of finding the lower-degree polynomial when we factor.
He used what would later be known as the " Ruffini - Horner method" to numerically approximate the root of a cubic equation.
InTartaglia did so only on the condition that Cardano would never reveal it and that if he did write a book about cubics, he would give Tartaglia time to publish.