Some quick thoughts about narrowing down a Question to make it better:
First you probably have some specifics in mind that weren't stated quite as explicitly as they could have been. For instance, "tell me all the identities that involve square roots on the left hand side" may have been intended to mean square roots $k^{1/2}$ of positive integers $k$. This begins the process of narrowing things down (and fits in with your example of the golden ratio).
Second, if pressed to narrow things further, you might decide to amplify (or discount) some of the familiar cases. For example, the Pythagorean theorem uses square roots. Is it of interest to you? E.g. "I know that $\sqrt{2}$ and $\sqrt{5}$ can appear as lengths of the hypotenuse of right triangles with integer length sides. For which integers $k$ is $\sqrt{k}$ possible?" The more specific the Question, the quicker Readers can determine if they know (or are interested enough to pursue) an Answer. Or, if something like that is not of interest, better to mention that it isn't than have Readers rushing to respond in that vein.
Finally, a brief mention of what motivates the Question is often helpful, if only to gauge the level of response that is appropriate. For example, you might say you have begun to read the Handbook of Mathematical Functions, and wondered if the early chapter on Elementary Analytic Methods omits some key identities for square roots. (Not that this was likely to be your motivation, just illustrating.) The description of what motivates the Question doesn't need to be lengthy; it just improves the Question to have a bit of context in formulating an apt Answer.
