Little old trick

I hope your job does not require you to be able to calculate squares quickly. Mine doesn't. But in case it does, here is a little trick for relatively small numbers.

First thing is to memorize squares upto 25. Schools generally hammer these into your brain, so that's a reasonable requirement I guess.

Suppose the number is between 25 and 50. Let's express it as 25+n, 25>n>0. Get the square of 25-n (now that you remember it). Take it and add 100n. Voila.
Let's take an example. Suppose you want to square 43.
43=25+n, n=18
25-18=7 whose square is 49. add 100n=18*100=1800 to it.
Answer is 1849.

Next step. Suppose the number lies between 50 and 100. Let's say 73.
73=50+n, n=23
So let's first find square of 50-n=27.
27 lies between 0 and 50 so you can use the trick mentioned above to find it if necessay.
Anyway it is 729.
Add to that 200n=4600. So 4600+729=5329, the required answer.
Note here we multiplied by 200 (as 'base' is 50, not 25), rest is same.

You can go on for x=100+n but then the amount of backtracking takes a little longer. And you have to multiply by 400.

The explanation of why this works is pretty simple, but it will be lengthy (in fact I started this post with it, but it just kept growing, and not to mention blogger is terrible with formatting), so I'll skip it.