Squaring Numbers in Range 26-50

Let A be such a number. Subtract 25 from A to get x. Subtract x from 25 to get, say, a. Then A² = a² + 100x. For example, if A = 26, then x = 1 and a = 25 - 1 = 24. Hence

26² = 24² + 100 = 676.

Similarly, if A = 37, then x = 37 - 25 = 12, and a = 25 - 12 = 13. Therefore,

37² = 13² + 100·12 = 1200 + 169 = 1369.

Why does this work?

Another way: recollect that (a ± b)² = a² ± 2ab + b². This leads to the following derivation:

Either way, it follows that

(25 + x)²