Question 283083
So, we want:

{{{x^5=x^4+x^3}}}

In other words we are asking when
{{{x^5-x^4-x^3=0}}}?
In even more words we are asking when
{{{x^3(x^2-x-1)=0}}}?
We note that x=0 certainly makes the statement true. Now, {{{x^2-x-1}}} will give two real (irrational) roots. So, strictly speaking there are three real numbers that meet the criterion: {0, 1/2 (1 - Sqrt[5]), 1/2 (1 + Sqrt[5]).

However, one should note that x=0 occurs as a root three times.