The following is part d.) of the project problem 50 from the chapter 1.1 (pp. 12) of Complex Analysis with Applications by Asmar.
Take it as given that $u^3v^3 = -\frac{p^3}{27}$ and $u^3 + v^3 = -q$ for real $u, v, p, q$, and that if two real numbers $U, V$ satisfy equations $U + V = -\beta, UV = \gamma$, then $U$ and $V$ are roots of the equation $X^2 + \beta X + \gamma = 0$. Therefore the previous $u^3$ and $v^3$ are roots of $X^2 + qX - \frac{p}{27} = 0$. With this we would like to conclude that $u = \sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}$ and $v = \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}$.
By using the equalities $v^3 = \frac{-p^3}{27u^3}$ and $v^3 + u^3 = -q$, it is quite straightforward to apply the quadratic equation to the equation $u^6 + qu^3 - \frac{p^3}{27} = 0$ to conclude that $u^3 = -\frac{q}{2} \pm \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}$. But I don't know how to proceed beyond this point. I mean that I need to argue why the $-$ in the $\pm$ has to disappear for $u$, and similarly why the $+$ needs to vanish for $v$. But I don't have any compelling argument for this. Sure, for a particular $u$ and $v$ the signs are as they need to be, but the problem statement gives away that these signs should be in general for the defined $u$ and $v$.
Edit: $u$ and $v$ are defined as $y = u + v$, where $y$ represent a change in variables by $x = y - \frac{a}{3}$, where the original cubic equation is $x^3 + ax^2 + bx + c = 0$ for real $a, b, c$.