<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] $$a-\sqrt{a+x}=x^2$$
$$a-x^2=\sqrt{a+x}$$
$$a+x=x^4-2ax^2+a^2$$
Consider it in terms of a quadratic in $a$
$$a^2+a(-2x^2-1)+x^4-x=0$$
Now we see that $x^4-x=x(x^3-1)=x(x-1)(x^2+x+1)=(x^2-x)(x^2+x+1)$. And $(x^2+x+1)+(x^2-x)=2x^2+1$. By Vieta, its roots are $x^2-x$ and $x^2+x+1$. You could also do it using quadratic formula.
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