Cardano's formula

Cardano’s formula (卡丹公式 / Cardano’s formula) is the closed-form solution for cubic equations, published by Gerolamo Cardano in 1545 in Ars Magna.

Setup. Any cubic $a{x}^3+b{x}^2+cx+d=0$ can be reduced (by substituting $x=t-\frac{b}{3a}$) to the depressed cubic:

$t^3+pt+q=0$

The formula. One root is:

$t=\sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$

The quantity under the square root, ${\Delta }^{\prime }=\frac{q^2}{4}+\frac{p^3}{27}$, acts like a discriminant:

• ${\Delta }^{\prime }>0$: one real root, two complex conjugate roots
• ${\Delta }^{\prime }=0$: repeated real roots
• ${\Delta }^{\prime }<0$: three distinct real roots — this is the famous casus irreducibilis, where the formula forces you through complex numbers even though all answers are real. This case is historically what pushed mathematicians to take $\sqrt{-1}$ seriously.

Quick example. $t^3-6t-9=0$, so $p=-6$, $q=-9$: ${\Delta }^{\prime }=\frac{81}{4}-8=\frac{49}{4}$, giving $t=\sqrt[3]{\frac{9}{2}+\frac{7}{2}}+\sqrt[3]{\frac{9}{2}-\frac{7}{2}}=\sqrt[3]{8}+\sqrt[3]{1}=3$. ✓

Derivation idea (the classic trick). Write $t=u+v$. Then $t^3=u^3+v^3+3uv(u+v)$, so the equation becomes $u^3+v^3+(3uv+p)t+q=0$. Impose $3uv=-p$; then $u^3+v^3=-q$ and $u^3{v}^3=-\frac{p^3}{27}$, so $u^3,v^3$ are roots of a quadratic — which you can solve. That’s the whole magic: a cubic collapses into a quadratic in $u^3$.

Fun history note since you like math lore: Cardano actually got the method from Tartaglia under an oath of secrecy, then published it anyway after discovering Scipione del Ferro had solved it earlier — one of math’s messiest priority fights. His student Ferrari then extended the idea to quartics, and Abel–Galois later proved degree ≥ 5 has no such general radical formula.