Before you understand this shortcut method, you are advice to read:
What are Square Numbers ?
Observer the following table:
Table 1
Natural Number  Square Number 

15  225  25  625  35  1225  45  2025 
Table one represents square of natural number having 5 at its one's place
You can observe that in square numbers:
Last two is always 25
Rest of the digits = [digit at ten's place X (1 + of digit at ten's place)]
Explaining the above observations with the help of following diagram again:
So now if you are asked to find square of 95 and 125, you can used the same technique and very quickly answer that:
95 = 9025
125 = 15625
(Elaboration of these again done in the following:)
