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. Sincerely, Patrick 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, where u = [ -q/2 + (q2/4 + p3/27)1/2]1/3 and 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. Doug Go to Math Central To return to the previous page use your browser's back button.