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[parent] Cardano's derivation of the cubic formula (Proof)

To solve the cubic polynomial equation $ x^3 + ax^2 + bx + c = 0$ for $ x$, the first step is to apply the Tchirnhaus transformation $ x = y-\frac{a}{3}$. This reduces the equation to $ y^3 + py + q = 0$, where

$\displaystyle p$ $\displaystyle =$ $\displaystyle b - \frac{a^2}{3}$  
$\displaystyle q$ $\displaystyle =$ $\displaystyle c - \frac{ab}{3} + \frac{2a^3}{27}$  

The next step is to substitute $ y = u-v$, to obtain
$\displaystyle (u-v)^3 + p(u-v) + q = 0$ (1)

or, with the terms collected,
$\displaystyle (q - (v^3 - u^3)) + (u-v)(p - 3 u v) = 0$ (2)

From equation (2), we see that if $ u$ and $ v$ are chosen so that $ q = v^3-u^3$ and $ p = 3uv$, then $ y = u-v$ will satisfy equation (1), and the cubic equation will be solved!

There remains the matter of solving $ q = v^3-u^3$ and $ p = 3uv$ for $ u$ and $ v$. From the second equation, we get $ v = p/(3u)$, and substituting this $ v$ into the first equation yields

$\displaystyle q = \frac{p^3}{(3u)^3} - u^3 $
which is a quadratic equation in $ u^3$. Solving for $ u^3$ using the quadratic formula, we get
$\displaystyle u^3$ $\displaystyle =$ $\displaystyle \frac{-27q + \sqrt{108 p^3 + 729 q^2}}{54} = \frac{-9q + \sqrt{12 p^3 + 81 q^2}}{18}$  
$\displaystyle v^3$ $\displaystyle =$ $\displaystyle \frac{27q + \sqrt{108 p^3 + 729 q^2}}{54} = \frac{9q + \sqrt{12 p^3 + 81 q^2}}{18}$  

Using these values for $ u$ and $ v$, you can back-substitute $ y=u-v$, $ p=b-a^2/3$, $ q=c-ab/3+2a^3/27$, and $ x = y-a/3$ to get the expression for the first root $ r_1$ in the cubic formula. The second and third roots $ r_2$ and $ r_3$ are obtained by performing synthetic division using $ r_1$, and using the quadratic formula on the remaining quadratic factor.



"Cardano's derivation of the cubic formula" is owned by djao.
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See Also: Ferrari-Cardano derivation of the quartic formula


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variant of Cardano's derivation (Proof) by mathcam
Cardano's formulae (Topic) by pahio
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Cross-references: factor, synthetic division, root, expression, quadratic formula, quadratic equation, cubic equation, terms, transformation, equation, polynomial
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This is version 8 of Cardano's derivation of the cubic formula, born on 2002-01-06, modified 2005-07-24.
Object id is 1408, canonical name is CardanosDerivationOfTheCubicFormula.
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AMS MSC: 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
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