@The-Blade-Dancer Thank you for writing and for being willing to share your comments. Looking back on this explanation, I sort of want to smack my head ... You're right, it is a bit obtuse. (Okay not a bit, but very!)

The question is asking whether $$a + b\sqrt{c} = x + y + 2\sqrt{xy}$$ for integers $$a, b,$$ and $$c$$ implies that $$x$$ and $$y$$ are always rational. (Rational means that the number is of the form $$\frac{m}{n}$$ where $$m$$ and $$n$$ are both integers.) Hmmm... it seems like this statement can't be true, just from using some common sense, because how can we be so lucky as to always get $$x$$ and $$y$$ rational... but how do we prove this?

Luckily, with a question of the form "It's always true that.... ", all we have to do is find a single counterexample where it isn't true.

I've updated the explanation to hopefully be a bit more simple and easy to understand. The counterexample I'm using is a case where $$x$$ and $$y$$ are both imaginary numbers but $$a,$$ $$b,$$ and $$c$$ are integers.