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 $(64)^{2}=(64+b)(64-b)+b^{2}$