<s>[INST] How to solve $x + \sqrt{x + \sqrt{11+x}}=11$ algebraically? How can I solve the following equation algebraically?
$x + \sqrt{x + \sqrt{11+x}}=11$
 [/INST] $x-11=-\sqrt{x+\sqrt{11+x}}$. Square both sides to get $(x-11)^2=x+\sqrt{11+x}$.
Then $(x-11)^2-x=\sqrt{11+x}$ and you square again
$((x-11)^2-x)^2=11+x$. Unfortunately you get a 4th order equation. Try this quartic equation
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