¿can anyone help me find the first root of the equation [math]-x^3+3

3,1801

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>>8661738
[math] x_0 = 3 [/math]
[eqn] x_{n+1} = x_n - \frac{-x_n^3+3x_n^2+\log(x_n+3)}{-3x_n^2+6x_n+\frac{1}{x_n+3}} [/eqn]
[math] x_n [/math] will converge to your root

The equation is satisphied when log(x+3) equals x^3-3x^2 so you just need to plot this two functions and see where is the interception

Is there any way to solve such an equation analytically?

>>8661976
define [math] i [/math] as the solution to this equation
congratulations, you now have created an analytic solution.
But for real, think about what you mean by analytical solution.
Do you mean in terms of simple functions?
What are usual functions? +,-,*,/,^,sin,cos,tan,arctan,erf? Where do you draw the line?

>>8661985
I think he meant a closed form or exact solution, not an approximation given by newton's method as per >>8661973

>>8661996
does x^2 - 2 = 0 have a closed form solution?
how do you think, a computer gives you the digits of sqrt(2), if not by approximation?

>>8661973
sorry i mean the root near -3, not 3, i tried with wolfram and it says that is -3 but then log(x+3) is undetermined

>>8661985
Pretty sure he means using elementary functions.

>>8663120
you're forgetting the minus, it's -x^3