Solve sqrt{a-sqrt{a+x}}=x

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=x^{2}+x+1\text{, }&\left(arg(x)<\pi \text{ or }x=0\right)\text{ and }arg(x+1)<\pi \\a=0\text{, }&x=0\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{4a+1}+1}{2}\text{, }&\left(arg(a-x^{2})<\pi \text{ and }arg(\frac{-\sqrt{4a+1}+1}{2})<\pi \text{ and }a\neq 0\right)\text{ or }a=0\\x=\frac{\sqrt{4a-3}-1}{2}\text{, }&\left(arg(a-x^{2})<\pi \text{ and }arg(\frac{\sqrt{4a-3}-1}{2})<\pi \text{ and }a\neq 1\right)\text{ or }\left(a=1\text{ and }arg(1-x^{2})<\pi \right)\\x=\frac{-\sqrt{4a-3}-1}{2}\text{, }&arg(a-x^{2})<\pi \text{ and }arg(\frac{-\sqrt{4a-3}-1}{2})<\pi \\x=\frac{\sqrt{4a+1}+1}{2}\text{, }&arg(a-x^{2})<\pi \text{ and }arg(\frac{\sqrt{4a+1}+1}{2})<\pi \end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=x^{2}+x+1\text{, }&\left(\frac{\sqrt{4x+1}+1}{2}>x^{2}\text{ or }x>0\right)\text{ and }x\geq 0\\a=x\left(x-1\right)\text{, }&\left(\frac{\sqrt{4x+1}+1}{2}>x^{2}\text{ and }x\geq 2\right)\text{ or }x=0\end{matrix}\right.$$