Bombelli’s Rule
Summary: Bombelli’s method (16th century) reduces any quartic equation to a cubic by expressing the quartic as a difference of two perfect squares; solving the auxiliary cubic then yields the four roots via two quadratics.
Sources: chapter-1.4.14
Last updated: 2026-05-03
Core Idea
Given , Bombelli rewrites it as:
so the quartic becomes a difference of two squares equated to zero. This factors immediately:
Two quadratics, each giving two roots.
The Auxiliary Cubic
Matching coefficients of the squared form to the original quartic forces (source: chapter-1.4.14, §767):
This is called the Bombelli cubic (or resolvent cubic). It has at least one real root (since every cubic does). Any one root suffices; all three give the same four quartic roots.
Once is known:
(If , use from the third condition.)
Relation to Bombelli’s Life and Cardan
Rafael Bombelli (c. 1526–1572) published this method in his Algebra (1572). Euler frames it as the natural next step after cardanos-rule: reducing degree by one. The principle—that solving degree presupposes solving degree —drives the entire Chapter XIV (source: §765).
Comparison with Euler’s Radical Ansatz
Both Bombelli and Euler’s method (Chapter XV, ch1.4.15-new-method-quartic) reduce the quartic to the same auxiliary cubic, though they arrive at it differently. Euler proves they yield identical roots (§769–770): all three roots of the Bombelli cubic produce the same four quartic roots.