Question

How to simplify this .?

`(4x^2 - y^2)/(12x^2 - 4xy-y^2)`

Answer 1

`(4x^2-y^2)/(12x^2-4xy-y^2) = (2x+y)/(6x+y)`

with exclusion `2x != y`

Explanation:

Given:

`(4x^2-y^2)/(12x^2-4xy-y^2)`

We can factor both numerator and denominator and then cancel any common factor.

The numerator is a difference of squares, so factors as:

`4x^2-y^2 = (2x)^2-y^2 = (2x-y)(2x+y)`

To factor the denominator find a pair of factors of `12` which differ by `4`. The pair `6, 2` works, so use that to split the middle term and factor by grouping:

`12x^2-4xy-y^2 = (12x^2-6xy)+(2xy-y^2)`

`color(white)(12x^2-4xy-y^2) = 6x(2x-y)+y(2x-y)`

`color(white)(12x^2-4xy-y^2) = (6x+y)(2x-y)`

So we find:

`(4x^2-y^2)/(12x^2-4xy-y^2) = (color(red)(cancel(color(black)((2x-y))))(2x+y))/((6x+y)color(red)(cancel(color(black)((2x-y)))))`

`color(white)((4x^2-y^2)/(12x^2-4xy-y^2)) = (2x+y)/(6x+y)`

with exclusion `2x != y`