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
+ px = q and
= 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
+ 16 = 12:r, while 4 is a true
solution of the equation
= 12x +
(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
Ars Magna, Chapter I.
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