Cardano and the quadratic formula

Problem

Compare the structure of Cardano’s formula with that of the $\Delta$ of the quadratic. What analogy stands out?

Solution

In the quadratic $ax^2+bx+c=0$ the solution is $x = \frac{-b\pm\sqrt{\Delta}}{2a}, \qquad \Delta=b^2-4ac,$ whereas in the depressed cubic $y^3+py+q=0$ Cardano’s formula is $y = \sqrt[3]{-\frac q2+\sqrt{D}}+\sqrt[3]{-\frac q2-\sqrt{D}}, \qquad D=\Bigl(\frac q2\Bigr)^2+\Bigl(\frac p3\Bigr)^3.$ The analogy is clear: in both cases a discriminant appears under a root ($\sqrt\Delta$ in the quadratic, $\sqrt D$ in the cubic) whose sign decides the number and nature of the real solutions. The “leap” is in the type of outer root: in the quadratic one takes a square root, in the cubic a cube root (indeed, the sum of two). Just as the quadratic requires reduction to normal form, the cubic first requires reduction to the depressed form. And it is here that, when $D<0$, the root of a negative number enters the scene: the formal analogy historically pushes towards the imaginary numbers.

Links

Topics: Quadratic equations
Concepts: Discriminant · Cubic discriminant · Cubic equation
Skills: Reasoning by cases
People: Gerolamo Cardano
Exercise type: Word problem