sqrt(a) + sqrt(b) = sqrt(x)

How many possible values of x less than or equal to 1000 are there?

Tristan's solution parallels the one I posted previously on many levels.  The main thing is that we both agree that the possible values of x are the number which are divisible by a perfect square greater than one.  Our final numerical answers differ by 1 though!

We have certainly used different counting methods, however!  While I chose to attempt counting the numbers that will work for x, he has counted the numbers that won't.  His counting method is interesting, and I think I understand how it works.  First he has counted all primes less than 1000.  Then, all numbers of the form 2p, 3p, 5p, 7p, ..., 29p, where p is prime (all of these numbers are composed, as he states, of 2 prime factors).  Then, all numbers of the form 2*3p, 2*5p, 2*7p, ..., 2*19p, and then 3*5p, 3*7p, ..., 3*17p, and then 5*7p and 5*11p.  Apparently, these are the only patterns for numbers less than 1000 that are composed of exactly 3 distinct prime factors.  Finally, he counts the numbers of the form 2*3*5p, 2*3*7p, 2*3*11p, and (I think?) 2*5*7p, which will all have 4 distinct prime factors.  (Obviously, any number with 5 or more distinct prime factors will by necessity be larger than 1000)

This is an ingenious solution!  However, on closer inspection I believe that I have found the discrepancy between Tristan's answer and mine.  About halfway through his counting, he lists

'5 between 13 and 39';

here he is counting numbers of the form 2*13p.  There are actually 6 primes that will work for this situation: 17, 19, 23, 29, 31, and 37.  Making this change will cause our final answers to agree.

There are 392 possible values for x.