The formula
Cardano's method, after reducing to the depressed cubic t³ + pt + q = 0
Why it works
Every cubic has at least one real root, because the graph runs from minus infinity to plus infinity and has to cross zero somewhere. Getting all of them means first removing the x² term with a substitution, which turns the equation into the depressed form Cardano's method can handle.
How to solve it by hand
- Divide through by a so the leading coefficient is 1
- Substitute x = t − b/3 to remove the t² term
- Apply Cardano's formula to the depressed cubic
- Undo the substitution to get back to x
What is worth knowing
The history here is worth knowing: Cardano published the method in 1545 after obtaining it from Tartaglia under an oath of secrecy, and the resulting feud is the most famous priority dispute in mathematics. The method also forced the first serious use of complex numbers — in the case with three real roots, Cardano's formula routes through square roots of negatives even though every answer is real. Mathematicians used them reluctantly for two centuries before accepting they were real objects. There is no equivalent formula beyond the fourth degree: Abel proved in 1824 that none can exist.