Q. 3 A5.0( 2 Votes )

Some natural numbers can be written as a difference of two perfect squares in two ways. For example.

24 = 72 – 52 = 52 – 12

32 = 92 – 72 = 62 – 22

40 = 112 – 92 = 72 – 32

Explain using algebra, the method of writing all multiples of 8, starting with 24 as the difference of two perfect squares in two ways.

Answer :

Let’s use algebra. starting with x,y, the square of the difference is is = + - 2

the square of the sum is


= + + 2


what if we subtract the square of the difference from the square of the sum.


- = (+ + 2 - (+ - 2)


- =


writing this in reverse


= -


for example, 24= 4


= here x=6 and y=1


= -


= -


= 72 – 52


similarly 24 can also be written as


24 = 432


= here x=3 and y=2


= -


= -


= 52 – 12


so 24 =72 – 52= 52 – 12


other multiple of 8,


32 = 92 – 72 = 62 – 22


32 = 481


= here x=8 and y=1


= -


= -


= 92 – 72


similarly 32 can also be written as


32 = 442


= here x=4 and y=2


= -


= -


32 = 92 – 72 = 62 – 22


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