Classification Guide

Types of Cubic Equations

Not all cubic equations behave the same way. Understanding the form helps you choose the fastest and clearest solving method.

General Cubic

ax³ + bx² + cx + d = 0

The most general form with all four coefficients. Normalize it before applying Cardano's method.

Monic Cubic

x³ + bx² + cx + d = 0

The leading coefficient is already 1, so normalization is complete.

Depressed Cubic

t³ + pt + q = 0

The quadratic term has been removed. This is the standard form used in Cardano's derivation.

Pure Cubic

x³ = k

Only the cubic and constant terms remain. The real root comes from a direct cube root.

Reducible Cubic

x³ - x = x(x² - 1) = 0

This cubic factors into lower-degree polynomials, so at least one root is visible by inspection.

Irreducible Cubic

x³ - 2 = 0

It does not factor over the rationals, so Cardano's method or a numerical method is required.

Comparison Table

Type	Quadratic Term	x Term	Best Method	Difficulty
General	Yes	Yes	Cardano	Hard
Monic	Yes	Yes	Cardano	Medium
Depressed	No	Yes	Cardano direct	Medium
Pure	No	No	Cube root	Easy
Reducible	Varies	Varies	Factoring	Easy
Irreducible	Varies	Varies	Cardano required	Hard

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Support & FAQ

Types of Cubics FAQ

Find quick answers to common questions about cubic equations and our solving methods.

Still have questions?

How do I know which type of cubic I have?

Check the coefficients. If a = 1, the cubic is monic. If the x^2 term is missing, it is depressed. If both x^2 and x are missing, it is pure.

Which type is easiest to solve?

Pure cubics are usually the easiest because they reduce to a cube-root step. Reducible cubics are also straightforward because factoring reveals the roots directly.

Can every cubic be converted to depressed form?

Yes. Any general cubic can be converted to t^3 + pt + q = 0 by substituting x = t - b/(3a).