The solution to the general cubic equation
The general cubic equation

1. x^3 + ax^2 + bx + c = 0 can be reduced to the form

2. y^3 + mx + n = 0.

by an appropriate substitution of x = y + z for some z in terms of the
coefficient a. The value z is suggested by the binomial expansion of  x = y + z in 1.

3. (y + z)^3 + a(y + z)^2 + b(y + z) + c = 0
    y^3 + 3y^2z + 3yz^2 + z^3 + ay^2 + 2ayz+az^2 + by + bz + c = 0

Collecting the y^2 or quadratic terms we have 3y^2z + ay^2.
If this expression were to equate to 0 and thus eliminate the quadratic term from 1., then
z = -a/3. So x = y-a/3 is the appropriate substitution to transform the general cubic in 1. to the
reduced form in 2.

We now are ready expand x = y – a/3 in 1. and solve for m and n in the reduced form in 2.
Using the binomial theorem again we have

4. (y – a/3)^3 + a(y –a/3)^2 + b(y –a/3) + c = 0.
    y^3 – 3ay^2/3 + 3ya^2/9 –a^3/27 + ay^2 - 2a^2y/3 + a^3/9 + by –ab/3 + c = 0.
    y^3 + 1/3(3b-a^2)y + 1/27(2a^3 – 9ab + 27c) = 0  Then
   m = 1/3(3b-a^2) and n = 1/27(2a^3 – 9ab + 27c)

At this point we have expressed m and n in 2. in terms of the known values a, b and c in 1.
We now must solve 2. for m and n to complete the solution. Notice that 2. can be easily solved
if we eliminate the linear term y. To reduce this equation to the more manageable form we
make another substitution y = w + v in 2

5. w^3 + v^3 = 0

for some w and v in terms of m. Similar to the argument we used to find z = -a/3, we expand
y = w + v in 2 to get,

 6. (w + v)^3 + m(w + v) + n = 0. w^3 + 3w^2v + 3wv^2 + v^3 + m(w + v) + n = 0. 

Now we collect the terms

3wv(w+v) + m(w+v)

Setting this to 0 gives us v = -m/3w as the appropriate value in y = w+v .
Expanding
y = w –m/3w in 2 we get 7. (w – m/3w)^3 + m(w - m/3w) + n = 0.
w^3 - 3w^2m/3w + 3wm^2/9w^2 - m^3/27w^3 + mw - m^2/3w + n = 0.
w^3 – wm + mw + m^2/3w – m^2/3w + m^3/27w^3 + n = 0
w^3 + m^3/27w^3 + n = 0 w^6 + n^w^3 = m^3/27
We complete the square by adding n^2/4 to both sides to get
     
     (w^3 + 1/2n)^2 = m^3/27 + n^2/4
                               ________________
 w^3 = -n/2 + -  \ / n^2/4 + m^3/27                     
                   ___________________________________
 w1  =  \3 /                   ________________
             \ /   -n/2 +  \ / n^2/4 + m^3/27                                
                   ___________________________________
 w2  =  \3 /                  ________________
             \ /  -n/2  -  \ / n^2/4 + m^3/27