Tartaglia's Solution of the Cubic and why it doesn'y give real values sometime?

In fact, it does give "real values' correctly when the cubic does have real roots. Tartaglia's method does frequently deliver messy answers that look like complex numbers, but in fact, when it is finally reduced into its simplest form (sometimes quite a feat to do so!), it ends up being a real number. A lot of math puzzles are based on this curiosity.