I was trying to solve the equation $x^3-2x^2-11x+12=0$ using Cardano's method, and I found myself with the following nested radical: $$\sqrt[3]{126i\sqrt{3}-55}$$
Is there any way to simplify this? I guess it has because I know from advance that this equation has nice solutions. Although, I cannot simplify it, even after researching on the subject. Most methods I tried take me to more nested radicals or more cubic equations.
Can somebody please help me?
Hint: let $\,a=\sqrt[3]{126i\sqrt{3}-55}\,$ and $\,b=\sqrt[3]{126i\sqrt{3}+55}\,$. Then:
$$ a^3 - b^3 = -110 $$ $$ ab = \sqrt[3]{-126^2 \cdot 3 - 55^2} = \sqrt[3]{-50653} = -37 $$
Writing $a^3-b^3 = (a-b)(a^2+ab+b^2) = (a-b)\big((a-b)^2 + 3ab)$ and letting $c=a-b$ gives:
$$c(c^2-3 \cdot 37) = -110$$
$$ c^3 - 111 c + 110 = 0$$
Factoring out the obvious root $c=1$ leaves a quadratic which gives the other two roots $\{-11,10\}$.
For each $c$, the values $a,b$ can be obtained by solving the quadratic with integer coefficients that results from $a-b=c\,, \;ab=-37\,$.