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The three roots $r_1, r_2, r_3$ of a cubic polynomial equation $x^3 + ax^2 + bx + c = 0$ are given by \begin{eqnarray*} r_1 & = & -\frac{a}{3} + \left(\frac{-2a^3 + 9ab - 27c + \sqrt{(2a^3-9ab+27c)^2 + 4(-a^2+3b)^3}}{54}\right)^{1/3} \\ & & {} + \left(\frac{-2a^3 + 9ab - 27c - \sqrt{(2a^3-9ab+27c)^2 + 4(-a^2+3b)^3}}{54}\right)^{1/3} \\ r_2 & = & -\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} \\ & & {} + \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} \\ r_3 & =
& -\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} \\ & & {} - \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} \end{eqnarray*}
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"cubic formula" is owned by djao.
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Cross-references: equation, polynomial, roots
There are 15 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 65150 times total.
Classification:
| AMS MSC: | 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros ) |
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Pending Errata and Addenda
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