Cardano's formula

Step 2: Cardano’s formula. Once the equation has been reduced to the depressed form $y^3+py+q=0$, the solution is obtained from a closed formula reminiscent of the one for the second degree.

Property — Cardano's formula for $y^3+py+q=0$

Setting the discriminant of the cubic $\Delta_3 = -4p^3-27q^2 \quad\text{(or equivalently }\, D=\Bigl(\tfrac{q}{2}\Bigr)^2+\Bigl(\tfrac{p}{3}\Bigr)^3\text{),}$ one solution is $y = \sqrt[3]{-\frac{q}{2}+\sqrt{D}}+\sqrt[3]{-\frac{q}{2}-\sqrt{D}}.$

The sign of $D$ plays the same role as the discriminant of the second degree: if $D>0$ there is a single real root, if $D<0$ there are three real ones (the “irreducible case”), if $D=0$ the real roots partly coincide.

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Topics: Second-degree equations
Concepts: Discriminant of the cubic · Third-degree equation · Depressed form
Methods: Cardano cubic
Skills: Using formulas
People: Gerolamo Cardano