December 9, 2022

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}= (2000)^{2}+2\times 2000\times 11+(11)^{2}

\, \, \, \, \, \, \, \, \, \, \, =4000000+ 44000+121

=4044121

Example : find the square of  1009.

Sol.  (1009)^{2}=(1000+9)^{2}= (1000)^{2}+2\times 1000\times 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 800\times 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  \, \, \, \, \, \, \, \, \, \, 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}= (507+b)(507-b)+b^{2}

Let b=7 . Then (507)^{2}= (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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