<s>[INST] Evaluate $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}$ Evaluate  $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}$
My attempt:
Let $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}=\sqrt{x}+\sqrt{y}$
Square both sides: $a+b+\sqrt{\left(2ab+b^2\right)}=x+2\sqrt{xy}+y$
Rearrange: $\sqrt{\left(2ab+b^2\right)}-2\sqrt{xy}=x+y-a-b$
That's where my lights go off.
Any leads? Thanks in advance.
 [/INST] After taking squares, you can proceed as follows
$$(a+b)+\sqrt{\left(2ab+b^2\right)}=(x+y)+2\sqrt{xy}$$
Compare corresponding (conjugate) parts on both the sides of above equation, we get
$$x+y=a+b\tag 1$$
$$2\sqrt{xy}=\sqrt{2ab+b^2}\iff 4xy=2ab+b^2\tag 2$$
$$x-y=\pm\sqrt{(x+y)^2-4xy}=\pm\sqrt{(a+b)^2-(2ab+b^2)}=\pm a\tag3$$
Solving (1) & (3) we get $x$ & $y$ as follows
$$x=a+\frac{b}{2}, \ y=\frac b2\ \ \ \text{OR}\ \  \ x=\frac b2, \ y=a+\frac{b}{2}$$
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