Working mod 100 we have that 44 is the only possibility for x^2 to have two digits. Which means this number AB (where A,B are digits) must end in 44. This means B=2 or B=8. Thus, 12,...,92 and 18,...,98. But since it is 4 digits the only possible choices are 32,...,92 and 38,...,98 and these are easy to check.