# Difference between revisions of "Cubic equation"

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An algebraic equation of degree three, i.e. an equation of the form | An algebraic equation of degree three, i.e. an equation of the form | ||

− | + | $$ax^3+bx^2+cx+d = 0$$ | |

+ | where $a\ne 0$. Replacing $x$ in this equation by the new unknown $y$ | ||

+ | defined by $x=y-b/2a$, one brings the equation to the following simpler | ||

+ | (canonical) form: | ||

+ | $$y^3+py+q = 0$$ | ||

+ | where | ||

+ | $$p=-\frac{b^2}{3a^2} + \frac{c}{a},$$ | ||

− | + | $$q=\frac{2b^3}{27a^3}-\frac{bc}{3a^2}+\frac{d}{a}$$ | |

+ | and the solution to this equation may be obtained by using | ||

+ | Cardano's formula (cf. | ||

+ | [[Cardano formula|Cardano formula]]); in other words, any cubic | ||

+ | equation is solvable in radicals. | ||

− | + | In Europe, the cubic equation was first solved in the 16th century. At the | |

+ | beginning of that century, S. Ferro solved the equation $x^3+px=q$, where | ||

+ | $p>0$, $q>0$, but did not publish his solution. N. Tartaglia rediscovered | ||

+ | Ferro's result; he also solved the equation $x^3+px+q$ ($(p>0$, $q>0$), and | ||

+ | announced without proof that the equation $x^3+q=px$ ($p>0$, $q>0$) could be | ||

+ | reduced to that form. Tartaglia communicated his results to | ||

+ | G. Cardano, who published the solution of the general cubic equation | ||

+ | in 1545. | ||

− | + | ====References==== | |

+ | <table><TR><TD valign="top">[1]</TD> | ||

+ | <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD> | ||

+ | </TR></table> | ||

− | |||

− | |||

− | + | ====Comments==== | |

− | + | The Eurpean history is treated in | |

− | + | [[#References|[a2]]], chap. 8. | |

− | + | In this book also results concerning cubic equations from | |

− | + | ancient Babylonia (2000 B.C.), ancient Chinese (Wang Hs'iao-t'ung, 625 | |

− | + | A.D.), and the most remarkable treatment of the cubic by the Persian | |

+ | mathematician Omar Khayyam (1024 -- 1123) are discussed. | ||

− | |||

− | |||

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> |

+ | <TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD> | ||

+ | </TR><TR><TD valign="top">[a2]</TD> | ||

+ | <TD valign="top">Howard Eves, "An introduction to the history of | ||

+ | mathematics", Saunders College Publishing, Philadelphia, 6. ed. (1964)</TD> | ||

+ | </TR></table> |

## Revision as of 18:46, 22 November 2011

An algebraic equation of degree three, i.e. an equation of the form

$$ax^3+bx^2+cx+d = 0$$ where $a\ne 0$. Replacing $x$ in this equation by the new unknown $y$ defined by $x=y-b/2a$, one brings the equation to the following simpler (canonical) form: $$y^3+py+q = 0$$ where $$p=-\frac{b^2}{3a^2} + \frac{c}{a},$$

$$q=\frac{2b^3}{27a^3}-\frac{bc}{3a^2}+\frac{d}{a}$$ and the solution to this equation may be obtained by using Cardano's formula (cf. Cardano formula); in other words, any cubic equation is solvable in radicals.

In Europe, the cubic equation was first solved in the 16th century. At the beginning of that century, S. Ferro solved the equation $x^3+px=q$, where $p>0$, $q>0$, but did not publish his solution. N. Tartaglia rediscovered Ferro's result; he also solved the equation $x^3+px+q$ ($(p>0$, $q>0$), and announced without proof that the equation $x^3+q=px$ ($p>0$, $q>0$) could be reduced to that form. Tartaglia communicated his results to G. Cardano, who published the solution of the general cubic equation in 1545.

#### References

[1] | A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) |

#### Comments

The Eurpean history is treated in [a2], chap. 8. In this book also results concerning cubic equations from ancient Babylonia (2000 B.C.), ancient Chinese (Wang Hs'iao-t'ung, 625 A.D.), and the most remarkable treatment of the cubic by the Persian mathematician Omar Khayyam (1024 -- 1123) are discussed.

#### References

[a1] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) |

[a2] | Howard Eves, "An introduction to the history of mathematics", Saunders College Publishing, Philadelphia, 6. ed. (1964) |

**How to Cite This Entry:**

Cubic equation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cubic_equation&oldid=15586