How can one solve the equation $\sqrt{x\sqrt{x} - x} = 1-x$? $$\sqrt{x\sqrt{x} - x} = 1-x$$
I know the solution but have no idea how to solve it analytically.

For the left hand side to be defined, you need $x\geq 1$ or $x=0$. Zero is not a solution. The left hand side will equal 0 for $x=1$ and will be strictly positive if $x>1$.
For $x\geq 1$, the right hand side is equal to 0 if $x=1$ and will be strictly negative if $x>1$. This shows that the only real solution is $x=1$.
