Help me solve x^3 + x + c = 0 equation

[amriths04]

x^3 + x + c = 0;

i need an expression for x. when i tried googling all those i could find was very messy and complicated. does anyone have a crisp solution( it need not be that accurate). it would be of great help to me.

 

[mmatica]

Re: finding the roots..

The (real) solution is
q - 1/(3*q)

where:
p = 12*(12 + 81 * c^2)^(1/2)
q = (p - 108*c)^(1/3) / 6

Is it too complicated?

 

[Dmitrij]

Re: finding the roots..

You've collided with the solution of the cubic algebraic equation. It's proved by Abel, that the roots can be found in radicals with use of complex values.

See Cardano formula in any book, devoted to linear algebra or just write "Cardano formula" in any search system (google,rambler, yandex, etc.)

If problems don't disappear, let me know. I'll write then this formula in the forum (I've not done it now, because it's too long, but if necessary, I'll do it, of course)

With respect.

Dmitrij

 

[echo47]

Re: finding the roots..

I typed this into MATLAB:
simplify(solve('x^3+x+c', 'x'))

It responded:
1/6*((-108*c+12*(12+81*c^2)^(1/2))^(2/3)-12)/(-108*c+12*(12+81*c^2)^(1/2))^(1/3)

1/12*(-(-108*c+12*(12+81*c^2)^(1/2))^(2/3)+12+i*3^(1/2)*(-108*c+12*(12+81*c^2)^(1/2))^(2/3)+12*i*3^(1/2))/(-108*c+12*(12+81*c^2)^(1/2))^(1/3)

-1/12*((-108*c+12*(12+81*c^2)^(1/2))^(2/3)-12+i*3^(1/2)*(-108*c+12*(12+81*c^2)^(1/2))^(2/3)+12*i*3^(1/2))/(-108*c+12*(12+81*c^2)^(1/2))^(1/3)