Cardano's Formula/Historical Note
Historical Note on Cardano's Formula
Cardano's Formula (in an incomplete form) was first published by Gerolamo Cardano in $1545$, in his Artis Magnae, Sive de Regulis Algebraicis. He learned the technique from Niccolò Fontana Tartaglia, who had sworn him to secrecy. However, as Cardano learned in $1543$, the technique had in fact first been discovered by Scipione del Ferro, so he no longer felt bound by his oath to Tartaglia. The latter did not see the matter in the same light, and entered into a feud with Cardano that lasted a decade.
The method detailed here is that given by Johannes van Waveren Hudde, who delivered it some time around $1650$.
This method was not actually analyzed in depth until the work of Rafael Bombelli, who was the first one to solve the problem of what to do about the "imaginary numbers" that inevitably arose when using this formula.