Lagrange resolvent for a cubic

For cubic roots ${\alpha }_i$ and a cube root of unity $\omega$, the Fourier combinations $u={\alpha }_1+\omega {\alpha }_2+{\omega }^2{\alpha }_3$ and $v$ have cubes satisfying a quadratic resolvent.

For $x^3+ax+b$, the Lagrange-resolvent cubes are roots of $X^2+27bX-27a^3$.

Cardano's formula solves a depressed cubic by taking square roots in its quadratic resolvent and then compatible cube roots.