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.
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 = 4
3
2
= here x=3 and y=2
= 
- 
= 
- 
= 52 – 12
so 24 =72 – 52= 52 – 12
other multiple of 8,
32 = 92 – 72 = 62 – 22
32 = 4
8
1
= here x=8 and y=1
= 
- 
= 
- 
= 92 – 72
similarly 32 can also be written as
32 = 4
4
2
= here x=4 and y=2
= 
- 
= 
- 
32 = 92 – 72 = 62 – 22