@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.