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The solution of cubic equation x^3+cx=d

Daniel Niu
Posted 3 years ago

The solution is complicated and even cannot understand.
Why can't Mathematica give a simple expression like this?
POSTED BY: Daniel Niu
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The version show the original post does not account for now-standard branch cuts, hence will not always evaluate numerically to a correct root.
POSTED BY: Daniel Lichtblau
Daniel Niu
Daniel Niu
Posted 3 years ago

Thank you for your reply, I am still confused that you mean the Cardano's formula is not generically correct? Could you please explain more? Your help would be highly appreciated.
POSTED BY: Daniel Niu

The result given has three solutions, all generically correct (that is, correct off a measure-0 set in the parameter space). The one proposed solution, in contrast, is not generically correct.
POSTED BY: Daniel Lichtblau

Another thing: Your formula is wrong for some values of the parameters c and d:

x^3 + 3 c x == 2 d /. 
   {x -> Power[d/2 + Sqrt[c^3/27 + d^2/4], (3)^-1] +
     Power[d/2 - Sqrt[c^3/27 + d^2/4], (3)^-1]} /.
  {c -> 2, d -> 1}

(*  False  *)
POSTED BY: Michael Rogers

(1) Don't you expect three solutions, not one? (Fundamental Theorem of Algebra.)

(2) Mathematica does have a command Simplify[], which helps.

(3) There will be some limitations on simplification because of branch cuts of the principal roots. If there are restrictions on c and d, including them in the assumptions of Simplify[] may help.
POSTED BY: Michael Rogers
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