# Definition:Cubic Equation

## Definition

A **cubic equation** is a polynomial equation of the form:

- $a x^3 + b x^2 + c x + d = 0$

### Discriminant

Let:

- $Q = \dfrac {3 a c - b^2} {9 a^2}$
- $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$

The **discriminant** of the cubic equation is given by:

- $D := Q^3 + R^2$

### Resolvent Equation

Let:

- $y = x + \dfrac b {3 a}$
- $Q = \dfrac {3 a c - b^2} {9 a^2}$
- $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$

Let $y = u + v$ where $u v = -Q$.

The **resolvent equation** of the cubic is given by:

- $u^6 - 2 R u^3 - Q^3$

## Also see

- Cardano's Formula: an algebraic technique for finding its roots.

- Results about
**cubic equations**can be found**here**.

## Historical Note

The ancient Greeks solved quadratics by geometric constructions, using the points of intersection of conic sections.

The general algebraic formulation of its solution remained unknown until as late as the end of the $15$th century, at which time Luca Bartolomeo de Pacioli wrote in his *Summa de Arithmetica* of $1494$ that the solution of the equations $x^3 + m x = n$ and $x^3 + n = m x$ were as impossible as Squaring the Circle.

However, this soon changed, as it was revealed shortly after that Scipione del Ferro was able to solve both of these, as well as the type $x^3 = m x + n$.

He passed this knowledge on to his apprentice Antonio Maria del Fiore.

Despite all attempts to keep the technique secret, it leaked out, and by $1535$ Niccolò Fontana Tartaglia also had the secret.

He was persuaded to tell Gerolamo Cardano, under an oath of secrecy.

However, Cardano learned that the secret had been obtained from Scipione del Ferro in the first place, so believed he was not bound by that oath, and so (to Tartaglia's annoyance) published it, in is *Artis Magnae, Sive de Regulis Algebraicis* of $1545$.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 9$: Solutions of Algebraic Equations: Cubic Equation - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Introduction - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**cubic** - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Where to begin... - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**cubic** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $4$: Lure of the Unknown: Cubic equations - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**cubic equation**