Date: Fri, 23 Apr 1999 09:47:13 +0200 (METDST)
Sender: Patrick
Subject: Question about 3rd degree polynomials

Dear Sir/Madam,
I found this E-mail address on your website, which I came upon after searching the web with the string "polynomials". I would like to know if you could help me with the following problem:

What is the general solution to the equation with the form:

a*x^3 + b*x^2 + c*x + d = 0

I have once seen a solution to this a few years ago, but I do not recall if it was a general solution. What I do know, is that you could simplify this equation to:

a*x'^3 + p*x' + q = 0

where x' = x - k, and k = -b/3a. p = 3*a*k^2 + 2*b*k + c and q = k^3 + b*k^2 + c*k + d

You can derive for what values of "a", "p", and "q" the equation has one or three (real) solutions. But I do not know how to proceed in order to find these solutions.

I would be very grateful if you could let me know the answer to this.

Delft, the Netherlands

Hi Patrick
Consider the cubic equation ax3 + bx2 + cx + d  = 0 in the variable x, where it assumed that a, b, c, and d are real coefficients. This equation has at least one solution x in the real numbers, but it could have exactly one real solution x and two solutions w and z that are complex numbers.
   To solve

ax3 + bx2 + cx + d  = 0

for a real number x, make the substitutions x' = x - k and k = -b/3a. In doing so, the original cubic equation now becomes
ax'3 + px' + q = 0.

where p = 3ak2 + 2bk + c and q = k3 + bk2 + ck + d
    To solve
ax'3 + px' + q = 0.

for x', one can use the method (formula) of Niccolo Fontano [or Tartaglia] (1499-1557), which was publicized in the book "Ars Magna" by Gerolamo Cardano (1501-1576). Clearly once x' is found, then the solution x to the original cubic is x = x' - k.
   The solution x' to
ax'3 + px' + q = 0.

is given by
x' = u - v,
u = [ -q/2 + (q2/4 + p3/27)1/2]1/3

v = [ q/2 + (q2/4 + p3/27)1/2]1/3

Because both u and v require the calculation of the square root of q2/4 + p3/27, the formula is most easily applicable to situations in which q2/4 + p3/27 is greater than or equal to 0.


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