Question 830950
We'll use numerical methods.

Graphing {{{x^3+3}}} and {{{x^2}}} together shows that there is only one point of intersection - slightly less than -1. So we know that there is only 1 real solution.

We use Newton's method:

{{{f(x)=x^3-x^2+3}}}

{{{df/dx=3x^2-2x}}}


then 

{{{x[n+1]=x[n]-(x[n]^3-x[n]^2+3)/(3x[n]^2-2x[n])}}}

We wont bother simplifying because we'll do the calculations by calculator:

guess x=-1

{{{x[2]=-6/5}}}
{{{x[3]=-47/40}}}
{{{x[4]=-1.1745595}}}