# Methods of Squaring

As we have seen in the case of multiplication, there are several methods for squaring also. Let us see the methods one by one.

# Method 1: Base 10 Method

Understand it by taking few examples:

- Â•Let us find out the square of 9. Since 9 is 1 less than 10, decrease it still further to 8. This is the left side of our answer.
- Â•On the right hand side put the square of the deficiency that is 1
^{2}. Hence, the answer is 81. - Â•Similarly, 8
^{2}= 64, 7^{2}= 49. - Â•For numbers above 10, instead of looking at the deficit we look at the surplus.

11^{2}= (11 + 1); 10 + 1^{2}= 121

12^{2}= (12 + 2); 10 + 2^{2}= 144

14^{2}= (14 + 4); 10 + 4^{2}= 18 ; 10 + 16 = 196

and so on.

This is based on the identities (

*a*+*b*) (*a*âˆ’*b*) =*a*^{2}âˆ’*b*^{2}and (*a*+*b*)^{2}=*a*^{2}+ 2*ab*+*b*^{2}.

# Method 2: Base 50n Method here, (n is any natural number)

This method is nothing but the application of (*a*+

*b*)

^{2}=

*a*

^{2}+ 2

*ab*+

*b*

^{2}.

Example-1

Find the square of 62.

Solution

Because this number is close to 50, we will assume 50 as the base.

(62)
To make it self explanatory a special method of writring is used.

^{2}= (50 + 12)^{2}= (50)^{2}+ 2 Ã— 50 Ã— 12 + (12)^{2 }= 2500 + 1200 + 144(62)

^{2}= [100â€™s in (Base)]^{2}+ Surplus | Surplus^{2}= 25 + 12 | 144 = 38 | 44 [Number before the bar on its left hand side is number of hundreds and on its right hand side are last two digits of the number.]

(68)

^{2}= 25 + 18 | 324 = 46 | 24(76)

^{2}= 25 + 26 | 676 = 57 | 76(42)

^{2}= 25 âˆ’ 8 | 64 = 17 | 24 [(*a*âˆ’*b*)^{2}=*a*^{2}âˆ’ 2*ab*+*b*^{2}]Example-2

Find the square of 112.

Solution

Since this number is closer to 100, we will take 100 as the base.

(112)

^{2}= (100 + 12)^{2}= (100)^{2}+ 2 Ã— 100 Ã— 12 + (12)^{2 }= 10000 + 2 Ã— 1200 + 144(112)

^{2}= [100â€™s in (Base)]^{2}+ 2 Ã— Surplus | Surplus^{2}= 100 + 2 Ã— 12 | 12
Alternatively, we can multiply it directly using base value method.

^{2}= 125 | 44^{2 }because assumed base here is 150.

(162)

^{2}= [100â€™s in (Base)]^{2}+ 3 Ã— Surplus | Surplus^{2}= 225 + 3 Ã— 12 | 12

^{2}= 262 | 44

# Method 3 : 10^{n} Method

This method is applied when the number is close to 10^{n}.

With base as 10

*, find the surplus or deficit (Ã—) Again answer can be arrived at in two parts*

^{n}(B + 2

*x*) |*x*^{2}^{}The right-hand part will consist of*n*digits. Add leading zeros or carry forward the extra to satisfy this condition.108

^{2}= (100 + 2 Ã— 8) | 8^{2}= 116 | 64 = 11664102

^{2}= (100 + 12 Ã— 2) | 2^{2}= 104 | 04 = 1040493

^{2}= (100 â€“ 2 Ã— 7) | (âˆ’7)^{2}= 86 | 49 â‡’ 86491006

^{2}= (1000 + 2 Ã— 6) | 6^{2}= 10|12 | 036 = 1012036
The right-hand part will consist of 2 digits. Add leading zeros or carry forward the extra to satisfy this condition.

63

^{2}= (25 + 13) | 13^{2}= 38 | 169 = 396938

^{2}= (25 â€“ 12) + (âˆ’12)^{2}= 13 | 144 = 1444
Square Mirrors14
^{2} + 87^{2} = 41^{2} + 78^{2}15
^{2} + 75^{2} = 51^{2} + 57^{2}17
^{2} + 84^{2} = 71^{2} + 48^{2}26
^{2} + 97^{2} = 62^{2} + 79^{2}27
^{2} + 96^{2} = 72^{2} + 69^{2} |

**Some Special Cases**- Numbers ending with 5

If a number is in the form of n5, the square of it is n (n+1) | 25

Example: 45

^{2}= 4 Ã— 5 | 25 = 2025135

^{2}= 13 Ã— 14 | 25 = 18225
This is nothing but the application of the multiplication method using the sum of units digits.

We can use this method to find out the squares fractions like â€¦also.

Process: Multiply the integral portion by the next higher integer and add.

For example,