1) Alex's house
2) Alex' house
When the noun ends with the letter 's' or 'x', do I need to put 's' after an apostrophe or not?
I remember I read some rules related to this in my school grammar book, but now I've forgotten it.
Use the regular apostrophe s: "Alex's" is correct.
Any name whose last syllable is pronunced with a long eez sound should have just the apostrophe, whereas others have apostrophe s: "Jones's", "Menzies's", "Kents's", "Jesus's", "Xerxes'" and "Euripides'" are correct.
Reference: Huddleston, R. Introduction to the Grammar of English, Cambridge University Press
I have never heard of an apostrophe following an
Yes, there is a rule saying that if somebody's name ends in 's' (not sure whether it is applicable to 'x' too), you can use either Charles' or Charles's and pronounce those forms accordingly - possessive apostrophes.
But to be on the safe side, I suggest using the Alex's form.
User114 is correct, but the explanation could be better. Use the 's if you add a vowel sound to the word to pronounce the possessive, whether or not the word is plural. If you say "Jones's" out loud, it has two syllables. If I had two young sons, I might refer to their shared bedroom as the kids' room.
But to expand further on these replies: -
If said aloud, it is immediately clear "my sons's bedroom" would have been incorrect because this would be pronounced: "my sonsez bedroom".
However, to the original point (but not original example), and this is where it gets less clear, both Charles' and Charles's are considered technically correct. Apparently though, Charles's is the "preffered" option, even though both would correctly be pronounced Charlesez. And from the other examples, apparently because Euripides' already ends with the "ez" sound, an additional s is seemingly not used; so why Menzies's, rather than Menzies'?
(Worse still, some countries apparently, USA in particular, don't necessarily consider (the language) English to be English (of we "Englanders") and wrongly try to take ownership. Then, if local rules vary, the above rules only strictly apply to "real" English. For example: American's officially misspell 'colour' as 'color'; use 'burglarized' instead of 'burgled'; and tragically, 'gotten' is legitimate grammar in the USA! As an Englishman I'd obviously argue that such changes are not correct English, even if "official" elsewhere. So in this regard, although I've never heard of the s being dropped after an x', strictly it's wrong regardless, even if possibly accepted in some places. Just try and change French, and beware the backlash!)
Also, note that in showing possesion, the 's is effectively used to contract "house of Alex", into "Alex's house"; just like o'clock is contracted from of the clock. So, the same principle used for a contraction is employed (like with: don't, you're, they'll, etc); except the phrase is usually reversed. This probably evolved in conversation, to help avoid unintentionally implying a plural (house's/houses); we don't pronounce the apostrophe after all.
This doesn't really explain why the s is introduced but it may help if you can simply remember: "of" or "of the" are replaced by the 's. If in doubt therefore, I'd recommend always using 's. So, always be consistent and use the "preferred" Charles's and avoid use of the confusing and "non-preferred" Charles' altogether. That way, hopefully, you should always be right.
So, certainly if you wish to write real English at least, only use Alex's.