I am getting really sick of these "mathema-tricks" postings, all from the same author, whose 50+ posts are all to do with the same web site.
The tl;dr: (50 + x)² = (25 + x) * 100 + x². If you know the squares from 0 to 25 you can then bootstrap the squares to 75. At the very least, between 41 and 59 you don't have to deal with carrying and can just say 42² = (25 - 8) 100 + 8² = 1764.
I would add that (100 + x)² = (100 + 2x) 100 + x², and doubling is efficient, too. So 92² = (100 - 16) 100 + 8² = 8464.
I'd do that kind of math as: 53ˆ2 = 50x50 + 3x50 + 3x50 + 3x3. No "trick" to remember (well perhaps that 5050 is like 55 plus that many zeroes at the end of the result)