<s>[INST] Solving $\sqrt{a-\sqrt{a+x}} = x$ Solve the equation
$$\sqrt{a-\sqrt{a+x}} = x$$
My approach:
Tried shifting the variables into different options, but couldn't get anything out of it. So, please help.
 [/INST] Let $\sqrt{a+x}=y\ge0\implies x=y^2-a$
$$y^2-a=\sqrt{a-y}\ge0\implies  y^2\ge a$$
$$(y^2-a)^2=a-y\iff a^2-(2y^2+1)a+y^4-y=0$$
$$a=\dfrac{2y^2+1\pm\sqrt{(2y^2+1)^2-4(y^4-y)}}2=\dfrac{2y^2+1\pm(2y+1)}2$$
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