8

1. Cubic Equations

In addition to the progress in calculation evidenced in Cardano's

Ars Magna, two fundamental developments are to be found that

would aid in the future development of mathematics, namely the

extension of the set of numbers to include first negative quantities

and then complex numbers. Cardano did not in fact use negative

numbers in the Ars Magna, which would have allowed him to solve

various types of cubic equations such as

x3

+ px = q and

x3

= px + q

as a single type. But he did show a greater openness to negative num-

bers by listing in addition to the "true" solutions to an equation the

negative solutions, which he called "false" solutions. For Cardano, a

"false" solution corresponded to a "true" solution of another equa-

tion, namely one with x replaced by —x. For example, for Cardano

—4 is a false solution to the equation

x3

+ 16 = 12:r, while 4 is a true

solution of the equation

x3

= 12x +

16.7

Exercises

(1) Find a solution to the cubic equation

x3 + 6x2 + 9x - 2 = 0.

(2) The cubic equation

x3 + 6x - 20 = 0

has 2 as a solution. How is this solution given by Cardano's

formula?

Ars Magna, Chapter I.