Shortcut for Square - A Simple Method

We have already discussed a method for finding the squares of the given number (check that method Here). Now we shall discuss another simplest method to calculate square of the given number. First of all one thing you should keep in mind, for you r convinience we've mentioned each and every step in detail. But You should do all these calculations MENTALLY. I mean without using pen or pencil. So practice well to do these problems orally. Now lets go to the topic, Finding the square of a number ending in 5 is very simple. If the last digit of the number is 5, then the last two digits of the square will be 25.What ever is the earlier part of the number multiply it with one more than itself and that will be the first part of the answer. (The second part of the answer will be 25 only).

35² = 1225
Here 3 x 4 = 12 so the answer is 1225

45²   =
2025

55²   =
3025

75²   =
5625

95²   =
9025

125²   =
15625

175²   =
30625

235²  =
55225

195²   =
38025

245²   =
60025

Well friends, now we know the squares of numbers 25, 30, 35, 40, 45, 50, 55 etc. If we want to find the squares of any other number, we find it using these squares which we already know.

For 262 we will go from 252, for 322 we go from 30 2 and so on.

One way is writing 262 = (25+1)2. But we need not even calculate (a+b)2 by adopting the following method.

262  = 252 + 26th odd number. i.e., 625 + 51 = 676

(a+b)2   = a2 + 2ab + b2

262   = (25 + 1 )2

           = 252  + 2 x 25 x 1 + 12

        = 625 + 50 + 1

       = 625 + 51 

        = 676

So, the answer is same. Now we shall look at an easiest method which will help you to calculate Squares Mentally. 

12   = 1 = 1

2 2  = 4 = 1+3

32   = 9 = 1+ 3 + 5

42   = 16 = 1 + 3 + 5 + 7

52   = 25 = 1 + 3 + 5 + 7 + 9

I mean, to get n2, you should add the FIRST n odd numbers. If we want 172, it will be the sum of the FIRST 17 odd numbers. 

n^th odd number is equal to (2n - 1)

Suppose you want to find out 62 knowing what 52 is, we can move from 52 to 62.

62 will be the sum of 1st 6 odd numbers. But the sum of the first 6 odd numbers can be written as " Sum of the first 5 odd numbers" + "Sixth odd number". Since we already know that the sum of the FIRST 5 odd numbers is 52 , i.e., 25, we need to add the sixth odd number, i mean (2x 6-1) = 11 to 25 to give us 62 = 36.

Similarly

312 = 900 + 31st Odd number = 900 + 61 = 961

362 = 1225 + 36th odd number = 1225 + 71 

                = 1296
(remember that 352 = 1225)

412 =  1600 + 81 = 1681

462 =  2025 + 91 = 2116

1262 = 15625 + 251 = 15876

1962 =  38025 + 391 = 38416

2162 =  46225 + 461 = 46656