Quote:
Originally Posted by jinydu
Ok, I'll try to frame my question more clearly this time.
Goal: Find a solution of x^3 + 6x  20 = 0 and express it in the simplest possible form.
Condition: Not allowed to use trialanderror guessing of rational roots or foreknowledge of the solution.
Hint: Applying Cardano's method gives:
x = cube root(10+sqrt(108)) + cube root(10sqrt(108)), but this may or may not be the simplest possible way of expressing this solution.

I'm not sure I understand the difficulty. It is easy to show
that 2 = x from x = cbr(10 + sqrt(108)) + cbr(10  sqrt(108)) = a + b
We have
a^3 + b^3 = 20
ab = 2
x = 2
x^3 = (a+b)^3 = 8
But (a+b)^3 = a^3 + b^3 + 3abx = 20  6x = 20  6*2 = 8 = 2^3
8 = 8 QED
I'm not sure what else you are looking for.