On p151 of my edition of Ian Stewart's "The Problems of Mathematics", he describes early work with imaginaries and Cardano's noting that Tartaglia's formula for solving a cubic, when applied to:
$$ x^3=15x+4 \tag{1}$$
yields
$$ x=\sqrt[3]{2}+\sqrt{-121}+ \sqrt[3]{2} - \sqrt{-121} \tag{2}$$
Bombelli, he then notes showed that:
$$ (2 \pm \sqrt{-1})^3 = 2 \pm \sqrt{-121} \tag{3}$$
Allowing us to restate (2) as:
$$ x=(2 + \sqrt{-1}) + (2 - \sqrt{-1}) = 4 \tag{4}$$
What I cannot follow is the substitution from (3) into (2) that gives us (4). Could someone explain?
Equation 2 should read $$x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}$$ which resolves the issue.