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cubic formula (Theorem)

The three roots $ r_1, r_2, r_3$ of a cubic polynomial equation $ x^3 + ax^2 + bx + c = 0$ are given by

$\displaystyle r_1$ $\displaystyle =$ $\displaystyle -\frac{a}{3} + \left(\frac{-2a^3 + 9ab - 27c + \sqrt{(2a^3-9ab+27c)^2 + 4(-a^2+3b)^3}}{54}\right)^{1/3}$  
    $\displaystyle {} + \left(\frac{-2a^3 + 9ab - 27c - \sqrt{(2a^3-9ab+27c)^2 + 4(-a^2+3b)^3}}{54}\right)^{1/3}$  
$\displaystyle r_2$ $\displaystyle =$ $\displaystyle -\frac{a}{3} - \frac{1+i\sqrt{3}}{2} \left(\frac{-2a^3 + 9ab - 27c + \sqrt{(2a^3-9ab+27c)^2 + 4(-a^2+3b)^3}}{54}\right)^{1/3}$  
    $\displaystyle {} + \frac{-1+i\sqrt{3}}{2} \left(\frac{-2a^3 + 9ab - 27c - \sqrt{(2a^3-9ab+27c)^2 + 4(-a^2+3b)^3}}{54}\right)^{1/3}$  
$\displaystyle r_3$ $\displaystyle =$ $\displaystyle -\frac{a}{3} + \frac{-1+i\sqrt{3}}{2} \left(\frac{-2a^3 + 9ab - 27c + \sqrt{(2a^3-9ab+27c)^2 + 4(-a^2+3b)^3}}{54}\right)^{1/3}$  
    $\displaystyle {} - \frac{1+i\sqrt{3}}{2} \left(\frac{-2a^3 + 9ab - 27c - \sqrt{(2a^3-9ab+27c)^2 + 4(-a^2+3b)^3}}{54}\right)^{1/3}$  



"cubic formula" is owned by djao.
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See Also: quartic formula, Galois-theoretic derivation of the quartic formula, Ferrari-Cardano derivation of the quartic formula, fundamental theorem of Galois theory

Other names:  cubic equation

Attachments:
Cardano's derivation of the cubic formula (Proof) by djao
Galois-theoretic derivation of the cubic formula (Proof) by djao
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Cross-references: equation, polynomial, roots
There are 13 references to this entry.

This is version 6 of cubic formula, born on 2002-01-06, modified 2005-03-05.
Object id is 1407, canonical name is CubicFormula.
Accessed 44657 times total.

Classification:
AMS MSC: 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
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undefined roots of a cubic? by abeckwith on 2004-05-14 14:34:32
Hello, PlanetMath.

I am a high school math teacher and designed a special project for my students that used the formula I found on your website for solutions to cubic equations. I also wrote a program to check their answers for a variety of coefficients, a, b, and c.

One student was assigned coefficients -3, 3, and 1. When she did her calculations, she ended up with an expression that involved dividing by 0, implying that the answer was undefined. I checked her calculations on my calculator program and discovered same. We looked at the graph of the cubic equation and found that it should have 1 real and 2 complex solutions. I also found that -3, 3, with any other value for the last constant would give a denom of 0.

My question is this: is there some restriction on the constants in the use of the formula that one should be aware of when using it? The page that has the formula mentions none.

Since this is a problem that has been solved completely, it would seem that the formula should produce answers for any values of the three constants.

Thank You.
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