Question

How do you add `\frac { a - 5} { a ^ { 2} + 1} + \frac { a + 1} { a ^ { 2} - 1}`?

Answer 1

`(2a^2-6a+6)/((a^2+1)(a-1))`

Explanation:

You add fractions by ensuring the denominators are the same. One way to get the same denominator is to multiply each fraction by the other denominator, like so

`(a-5)/(a^2+1) xx (color(red)(a^2-1))/(color(red)(a^2-1))+(a+1)/(a^2-1)xx(color(blue)(a^2+1))/(color(blue)(a^2+1))`

With identical denominators, we can now add the numerators

`((a-5)(a^2-1)+(a+1)(a^2+1))/((a^2+1)(a^2-1))`

It may be useful to factor out `(a^2-1)`, giving

`((a-5)(a-1)(a+1)+(a+1)(a^2+1))/((a^2+1)(a-1)(a+1))`

This allows us to cancel the `(a+1)` terms

`((a-5)(a-1)cancel((a+1))+cancel((a+1))(a^2+1))/((a^2+1)(a-1)cancel((a+1)))`

Expanding the numerator gives

`(a^2-6a+5+a^2+1)/((a^2+1)(a-1))`

Combing like terms in the numerator

`(2a^2-6a+6)/((a^2+1)(a-1))`