If $a+\sqrt{a}=b+\sqrt{b}$ is $a=b$? If $a+\sqrt{a}=b+\sqrt{b}$, does this automatically mean that $a=b$?
I first tried to square both sides but that seemed to get me nowhere.
$$a-b=\sqrt{b}-\sqrt{a}$$
Can we just conclude that $a$ has to be equal to $b$ to make this expression to be true?

$a-b=\sqrt{a}-\sqrt{b} \implies (\sqrt{a}-\sqrt{b})(-1+\sqrt{a}+\sqrt{b})=0\implies \sqrt{a}=\sqrt{b}$  or $1=\sqrt{a}+\sqrt{b}$. So it might not be the case that $a = b$.
I realize my answer above is for a different problem. So back to this one. We have $a - b = \sqrt{b} - \sqrt{a} \implies a - b +\sqrt{a}-\sqrt{b} = 0 \implies (\sqrt{a}-\sqrt{b})(1+\sqrt{a}+\sqrt{b}) = 0 \implies \sqrt{a} = \sqrt{b} \implies a = b$ .
