September 14, 2024

Squaring of numbers using formulae

Squaring of numbers  means multiplying a number by itself. There are various formulae used in general mathematics  to square numbers instantly. Let us discuss them one by one.

(i) $(a+b)^{2}=a^{2}+2ab+b^{2}$

This formula is generally use  to square  those numbers which are near multiples of 10.  For example ,  suppose we have to find the square of  2011.  We represent the number 2011 as 2000+11. Thus we have converted it into a form of (a+b) where the value of a is 2000 and the value of b is 11.

Now $(2011)^{2}=(2000+11)^{2}=&space;(2000)^{2}+2\times&space;2000\times&space;11+(11)^{2}$

$\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;=4000000+&space;44000+121$

=4044121

Example : find the square of  1009.

Sol.  $(1009)^{2}=(1000+9)^{2}=&space;(1000)^{2}+2\times&space;1000\times&space;9+(9)^{2}$

= 1000000+18000+81

=1018081

(ii) $(a-b)^{2}=a^{2}-2ab+b^{2}$

This formula is very much like the first one. The only difference is that the middle term contain negaive sign.

Example : Find the square of 792.

Sol.  $(792)^{2}=(800-8)^{2}=(800)^{2}-2\times&space;800\times&space;8+(8)^{2}$

=640000-  12800+64

=6 27264.

(iii)  $a^{2}=(a+b)(a-b)+b^{2}$

( How it comes!  We know that    $a^{2}-b^{2}=(a+b)(a-b)$

. Therefore  $\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;\,&space;a^{2}=(a+b)(a-b)+b^{2}$   )

Suppose we are asked to find the square of a number. Let’s call this number ‘a’. Now in this case we will find another number  ‘b’ in such a way that  the product  (a+b)(a-b)  and the square of ‘b’ can be easily find.

Example 1.  Find the square of 66.

Sol. In this case, the value of  ‘a’ is 64. Now, we know that

$a^{2}=(a+b)(a-b)+b^{2}$

substituting   the value of ‘a’  as 64 , we get     $(64)^{2}=(64+b)(64-b)+b^{2}$

Now , we have to  substitute  the value of ‘b’ with  such a number that the whole equation becomes easy to solve. Let us suppose  the value of b= 4.

Then the  equation becomes    $(64)^{2}=(64+4)(64-4)+4^{2}$

=68 x 60 +16

= 4080+16

= 4096

Example 2.   Find the square of 507.

$(507)^{2}=&space;(507+b)(507-b)+b^{2}$

Let b=7 . Then $(507)^{2}=&space;(507+7)(507-7)+7^{2}$

= 514 x 500+ 49

= 257000+49

= 257049

Thus we see that  the above formulae can help us to find the squares of any number above and below a round figure respectively.

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