24 August 2021

Solving the cubic and quartic equations

All the numbers used here are complex. What follows is elementary and self-contained; it requires nothing beyond finding roots of complex numbers, i.e., de Moivre's formula.

The cubic


The equation to be solved is (1). The new variable defined in (2) does away with the quadratic term, and we are left with (3), with coefficients 3p and 2q chosen for convenience.



The quartic


The equation to be solved is (4). The new variable defined in (5) does away with the cubic term, and we are left with (6), with a coefficient 2q chosen for convenience. The following theorem shows how to reduce (6) to one cubic and two quadratic equations.