Question

How do you simplify `(x+5 )/ (x^2-25)`?

Answer 1

`1/(x-5)`

Explanation:

`x^2-25" is a "color(blue)"difference of squares"`

`a^2-b^2=(a-b)(a+b)`

`x^2-25=x^2-5^2=(x-5)(x+5)`

`(cancel((x+5)))/(cancel((x+5))(x-5))=1/(x-5)`

`"with restriction "x!=5`

Answer 2

`1/(x-5)`

Explanation:

The key realization is that our denominator fits the difference of squares pattern `a^2-b^2`, which factors as `(a+b)(a-b)`.

We can rewrite `x^2-25` as `(x+5)(x-5)`, which allows us to rewrite our original expression as

`(x+5)/((x+5)(x-5))`

The `x+5` terms on the top and bottom cancel, and we're left with

`cancel(x+5)/(cancel(x+5)(x-5))`

`1/(x-5)`

Hope this helps!