# cubic equation

A cubic equation is a polynomial equation of the third degree, the general form of which is

*ax*^{3} + *bx*^{2} + *cx* + *d* =
0,

where *a*, *b*, *c*, and *d* are constants.

There was a great controversy in 16th-century Italy between Girolamo Cardano and Niccoló Tartaglia about who should
get credit for solving the cubic. At this time symbolic algebra hadn't been
developed, so all the equations were written in words instead of symbols.
Early studies of cubics helped legitimize negative
numbers, give a deeper insight into equations in general, and stimulate
work that eventually led to the discovery and acceptance of complex
numbers. Cardano, in his *Ars magna*, found negative solutions
to equations, but called them "fictitious". He also noted an important fact
connecting solutions of a cubic equation to its coefficients,
namely, that the sum of the solutions is the negation of *b*, the coefficient
of the *x*^{2} term. At one other point, he mentions that the
problem of dividing 10 into two parts so that their product is 40 would
have to be 5 + *v*(–15) and 5 – *v*(–15). Cardano didn't go further
than this observation of what later came to be called complex numbers, but
a few years later Rafael Bombelli (1526–1672) gave several examples
that involved these strange new mathematical beasts.