Name: Factorisation Using Difference of Two Squares
Uploaded: 2019-07-19
Duration: 5 min 14 s
Description: Sometimes writing a polynomial as the difference of two squares is the only way to factorise it. So let’s learn how to do it in this simple video!

Question

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

$24 = 7^{2} - 5^{2} = 5^{2} - 1^{2}$

$32 = 9^{2} - 7^{2} = 6^{2} - 2^{2}$

$40 = 11^{2} - 9^{2} = 7^{2} - 3^{2}$

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

Accepted Answer

Let’s use the identity of the square of the difference is $24 = 7^{2} - 5^{2} = 5^{2} - 1^{2}$

The square of the sum is

$\Rightarrow (x + y)^{2} = x^{2} + y^{2} + 2 x y$

Now, subtract the square of the difference from the square of the sum.

$\Rightarrow (x + y)^{2} - (x - y)^{2} = (x^{2} + y^{2} + 2 x y) - (x^{2} + y^{2} - 2 x y)$

$\Rightarrow (x + y)^{2} - (x - y)^{2} = 4 x y$

$\Rightarrow 4 x y = (x + y)^{2} - (x - y)^{2}$

For example, $24 = 4 \times 6 \times 1$

Here, $x = 6$ and $y = 1$

$\Rightarrow (x + y)^{2} - (x - y)^{2}$

$\Rightarrow (6 + 1)^{2} - (6 - 1)^{2}$

$\Rightarrow 7^{2} - 5^{2}$

Similarly, $24$ can also be written as

$24 = 4 \times 3 \times 2$

Here, $x = 3$ and $y = 2$

$\Rightarrow (x + y)^{2} - (x - y)^{2}$

$\Rightarrow (3 + 2)^{2} - (3 - 2)^{2}$

$\Rightarrow 5^{2} - 1^{2}$

Therefore,

$\Rightarrow 24 = 7^{2} - 5^{2} = 5^{2} - 1^{2}$

Other multiples of $8$ ,

$\Rightarrow 32 = 9^{2} - 7^{2} = 6^{2} - 2^{2}$

$\Rightarrow 32 = 4 \times 8 \times 1$

Here, $x = 8$ and $y = 1$

$\Rightarrow (x + y)^{2} - (x - y)^{2}$

$\Rightarrow (8 + 1)^{2} - (8 - 1)^{2}$

$\Rightarrow 9^{2} - 7^{2}$

Similarly, $32$ can also be written as $32 = 4 \times 4 \times 2$

Here, $x = 4$ and $y = 2$

$\Rightarrow (x + y)^{2} - (x - y)^{2}$

$\Rightarrow (4 + 2)^{2} - (4 - 2)^{2}$

$\Rightarrow 6^{2} - 2^{2}$

Important Questions on Identities

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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.

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