# Difference between revisions of "Cubic equation"

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 $x=y-b/3a$, 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)