How to solve cube roots question ?

[sunny86]

How to solve cube roots question ?
Example :
x^3 - 100x^2 - 7800x + 16300 = 0

I had think long time but still cannot find the way. Besides trial an error, is there anyway to solve this problem ?
thank you.

 

[WigneRacah]

There are numerical methods to get approximate solutions ...

For an exact solution you may find useful this

http://mizar.uwb.edu.pl/JFM/Vol12/polyeq_1.html"

 

[CRGreathouse]

Cardano had a method, but you might find it a bit lengthy for your taste. Here's MathWorld's page on cubics:

http://mathworld.wolfram.com/CubicFormula.html

 

[radou]

For a given equation $$x^3+ax^2+bx+c=0$$, we can substitute $$x=y-\frac{a}{3}$$, which implies $$y^3+py+q=0$$, where $$p=b-\frac{a^2}{3}$$, and $$q=\frac{2a^3}{17}-\frac{ab}{3}+c$$. After further manipulation, one can retrieve $$x=\sqrt[3]{-\frac{q}{2}+\sqrt{(\frac{q}{2})^2+(\frac{p}{3})^3}}+\sqrt[3]{-\frac{q}{2}-\sqrt{(\frac{q}{2})^2+(\frac{p}{3})^3}}$$, which represents Cardan's formula.

 

[sunny86]

thank guy ~