Solve for $k$ in the equation ${\sqrt{k + \sqrt{x-k}}} -x = 0$

Question

Solve for $k$ in the equation ${\sqrt{k + \sqrt{x-k}}} -x = 0$

I could only try standard methods so far.

${\sqrt{k - \sqrt{x+k}}} = x$

$({\sqrt{k - \sqrt{x+k}}})^2 = x^2$

${{k - \sqrt{x+k}}} = x^2$

${{k - \sqrt{x+k}}} = x^2$

${\sqrt{x+k}} = k - x^2$

$ x + k = k^2 - 2kx^2 + x^4$

$ k^2 - 2kx^2 + x^4-x-k = 0$

That's as far as I got. Any suggestions on how to isolate $k$ here?

Answer 1

We can separate this into

$$k^2-k(2x^2-1)+(x^4-x) = 0$$

and use the quadratic formula to get

$$k = \frac{2x^2-1\pm\sqrt{-4x^2+4x+1}}{2}$$

One can check to see which of these possible solutions work, although that is fairly cumbersome.