Cardano in action

Let us apply Cardano’s formula to a cubic already in depressed form.

Example — Cardano in action: $y^3+6y-20=0$

Here $p=6$, $q=-20$. $D = 100 + 8 = 108 > 0$ (a single real root). $y = \sqrt[3]{10+\sqrt{108}}+\sqrt[3]{10-\sqrt{108}} = \sqrt[3]{10+6\sqrt{3}}+\sqrt[3]{10-6\sqrt{3}} = 2.$ The identity $(1+\sqrt{3})^3 = 10+6\sqrt{3}$ lets us simplify the cube roots.

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Topics: Second-degree equations
Concepts: Third-degree equation
Methods: Cardano cubic
Skills: Solving equations · Using formulas