TITLE: How can one solve the equation $\sqrt{x\sqrt{x} - x} = 1-x$?
QUESTION [10 upvotes]: $$\sqrt{x\sqrt{x} - x} = 1-x$$
I know the solution but have no idea how to solve it analytically.

REPLY [5 votes]: The equation
$$\sqrt{x\sqrt{x}-x} = 1-x$$
implies that both sides,
and the argument of each square root,
are all nonnegative.
Thus any solution must obey
$$\sqrt{x}\text{ exists}\implies x\ge0$$
$$0\le1-x\implies x\le1$$
$$0\le x\sqrt{x}-x=x\left(\sqrt{x}-1\right)
\implies \sqrt{x}\ge1
\implies x\ge1$$
But then $1\le x\le 1 \implies x=1$ is the only solution.