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| ``Re: undefined roots of a cubic?''
by TomLK on 2004-05-16 18:26:39 |
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I do not know which "formula" for solving the cubic equation you have,
but the one first found by Scipione del Ferro (beginning of 16. century), popularized by Girolamo Cardano, and subsequently known as Cardano's, is valid for any cubic equation
z^3+az^2+cz+d=0.
Introduction of a new variable via z=y-a/3 converts the equation into the "canonic" form
y^3+py+q=0.
By a most ingenious argumnet Del Ferro found that its solutions are given by the formula
y=t^{1/3}+s^{1/3}
where the two terms on the right side are any of three (among nine) pairs of third roots of
t=-q/2+[(q/2)^2+(p/3)^3]^{1/2},
s=-q/2-[(q/2)^2-(p/3)^2]^{1/2},
whose product is equal to -p/3.
For a fine historical account, and del Ferro's procedure, see, e.g.,
J. Dieudonne's "Pour l'honneur de l'espirit humain", Hachette, Paris (1987).
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