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Re: cubic complex

  • To: mathgroup at smc.vnet.net
  • Subject: [mg30748] Re: [mg30728] cubic complex
  • From: BobHanlon at aol.com
  • Date: Sun, 9 Sep 2001 03:26:43 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

In a message dated 2001/9/8 3:12:28 AM, dmerdol at mech.ubc.ca writes:

>I am after the analytical solution for
>      a0 * x^3 + a1 * x^2 + a2 * x + a3 = 0 .
> The worst thing about this equation is the coefficients 
>(ai), because they complex numbers.
>

Solve[a0 * x^3 + a1 * x^2 + a2 * x + a3 == 0, x]//Simplify

{{x -> (1/(6*a0))*(-2*a1 + 2^(2/3)*
       (-2*a1^3 + 9*a0*a2*a1 - 27*a0^2*a3 + 
         Sqrt[(2*a1^3 - 9*a0*a2*a1 + 27*a0^2*a3)^2 - 
           4*(a1^2 - 3*a0*a2)^3])^(1/3) + 
      (2*2^(1/3)*(a1^2 - 3*a0*a2))/
       (-2*a1^3 + 9*a0*a2*a1 - 27*a0^2*a3 + 
         Sqrt[(2*a1^3 - 9*a0*a2*a1 + 27*a0^2*a3)^2 - 
           4*(a1^2 - 3*a0*a2)^3])^(1/3))}, 
  {x -> (1/(12*a0))*(-4*a1 + I*2^(2/3)*(I + Sqrt[3])*
       (-2*a1^3 + 9*a0*a2*a1 - 27*a0^2*a3 + 
         Sqrt[(2*a1^3 - 9*a0*a2*a1 + 27*a0^2*a3)^2 - 
           4*(a1^2 - 3*a0*a2)^3])^(1/3) - 
      (2*I*2^(1/3)*(-I + Sqrt[3])*(a1^2 - 3*a0*a2))/
       (-2*a1^3 + 9*a0*a2*a1 - 27*a0^2*a3 + 
         Sqrt[(2*a1^3 - 9*a0*a2*a1 + 27*a0^2*a3)^2 - 
           4*(a1^2 - 3*a0*a2)^3])^(1/3))}, 
  {x -> -((1/(12*a0))*(4*a1 + 2^(2/3)*(1 + I*Sqrt[3])*
        (-2*a1^3 + 9*a0*a2*a1 - 27*a0^2*a3 + 
          Sqrt[(2*a1^3 - 9*a0*a2*a1 + 27*a0^2*a3)^2 - 
            4*(a1^2 - 3*a0*a2)^3])^(1/3) + 
       (2*2^(1/3)*(1 - I*Sqrt[3])*(a1^2 - 3*a0*a2))/
        (-2*a1^3 + 9*a0*a2*a1 - 27*a0^2*a3 + 
          Sqrt[(2*a1^3 - 9*a0*a2*a1 + 27*a0^2*a3)^2 - 
            4*(a1^2 - 3*a0*a2)^3])^(1/3)))}}


Bob Hanlon
Chantilly, VA  USA


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