The number of real solutions of complicated surd

Question

I have tried to solve this question, and I found the one real root, x=0, but I do not understand how to show that there is only one root: Find the number of real solutions of $x+\sqrt{x^2+\sqrt{x^3+1}}=1$
The solution to the question, is 1 real solution, but I am not sure how to prove that there are no other solutions.

Answer 1

For the real square root to be defined $\,\sqrt{x^3+1}\,$, it is necessary that $\,x^3+1 \ge 0 \iff x \ge -1\,$. For $\,\sqrt{x^2+\sqrt{x^3+1}}=1-x \ge 0\,$, it is necessary that $\,x \le 1\,$, so any real solution is in $\,[-1,1]\,$.

Eliminating the radicals:

$$ x^3+1 = \left((1-x)^2-x^2\right)^2 \;\;\iff\;\; x^3 - 4 x^2 + 4 x = 0 \;\;\iff\;\; x(x-2)^2=0 $$

The latter has the unique real solution $\,x=0\,$ in $\,[-1,1]\,$.