How to find x from the equation $\sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} = m \sqrt{\frac{x}{x+\sqrt{x}}}$ For m are real number,
Find x from the equation
$$\sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} = m \sqrt{\frac{x}{x+\sqrt{x}}}$$
I tried to multiply $\sqrt{x + \sqrt{x}}$ to the both sides and I get
$$x + \sqrt{x} - \sqrt{x^{2} - x} = m \sqrt{x}$$
What should I do now to get x? Can anyone show me a hint please?

After dividing by $\sqrt{x}$ you may use the substitution $\boxed{y = \sqrt{x}}$. Then, after a bit rearranging you get
$$y+1 - m = \sqrt{y^2-1}$$
Squaring removes the $y^2$ and you can solve directly for $y$:
$$y = \frac{2-2m+m^2}{2(m-1)} \Rightarrow \boxed{x = \frac{(2-2m+m^2)^2}{4(m-1)^2}}$$
Note, that there are solutions only for $m >1$, since $y\geq 0$.
