What's a general way to square any two-digit number without a special rule?

Use the cross method on the number times itself — units = b², middle = 2·a·b, hundreds = a² — then carry.

For ab, squaring gives columns a² | 2ab | b²:

47² → units 7² = 49 → write 9, carry 4 → middle 2·4·7 = 56, +4 = 60 → write 0, carry 6 → hundreds 4² = 16, +6 = 22 → 2209.

Check: 47² = 2209 ✓. It's the (10a+b)² expansion. This always works, so it's your fallback when a number isn't near 50, near 100, or ending in 5.