Question

factorise give full solution

Answer 1

Hello,
${x}^{2} + \frac{1}{ {x}^{2} } - 2 - {y}^{2}$

Lets first group.
It will be easier to understand.

Also,
Remember the formulae that will be used in the answer.

${(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy \\ \\ {x}^{2} - {y}^{2} = (x + y)(x - y)$
These are also known as algebraic identities.

Now,
Lets come to the question.

${x}^{2} + \frac{1}{ {x}^{2} } - 2 - {y}^{2} \\ = ( {x}^{2} + \frac{1}{ {x}^{2} } - 2) - {y}^{2} \\ \\ = {((x)}^{2} + ( { \frac{1}{x}) }^{2} - 2.x. \frac{1}{x} ) - {y}^{2}$
Here we can see the
First identity mentioned above.
So,
Accordingly making it into one square we get

$= ( { x - \frac{1}{x} )}^{2} - {y}^{2}$
Now,
On looking carefully we find the second identity.
So,
Factorizing accordingly.

$= ( {x - \frac{1}{x} )}^{2} - ( {y)}^{2} \\ \\ = (x - \frac{1}{x} + y)(x - \frac{1}{x} \: - y)$
So,

Now, the factorization is complete and the final answer is
$= (x - \frac{1}{x} + y)(x - \frac{1}{x} \: - y)$

Hope this will be helping you ✌️

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