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.

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