Solving a system of 3 nonlinear equations

EM_Guy

a = xyz
b = xy+xz+yz
c = x + y + z

How do you solve x, y, and z?

mfb

x=c-y-z
b=(c-y-z)y + (c-y-z)z + yz = c(y+z)-y^2-zy-z^2
Solve this quadratic equation for y (or z), use both in a=xyz and hope that it has a nice solution?

EM_Guy

It is not a quadratic equation. And it is not a "nice" solution.

I have determined that z^3-cz^2+bz-a = 0. So, if we can find the roots of the cubic function, then we have z as a function of a, b, and c. Then, it should be straightforward to find x and y in terms of a, b, and c.

But I forget how to find the roots of a cubic function.

mfb

b= c(y+z)-y^2-zy-z^2 is a quadratic equation in y (or z).

Solutions of a cubic equation
z^3-cz^2+bz-a = 0 looks very nice in my opinion.