Question

How do you evaluate `\frac { 4} { x ^ { 2} - 25} + \frac { 6} { x ^ { 2} + 6x + 5}`?

Answer 1

1. Factor the denominators
2. Find LCD
3. Make equivalent fractions with Like denominators
4. Add like Terms

`(10x - 26)/((x-5)(x+5)(x+1) )`

Explanation:

`4/(x^2 -25) + 6/(x^2 + 6x +5)`

to factor `x^2-25` you need to understand difference of squares where `(x-5)(x+5) = x^2 -25`

`4/((x-5)(x+5)) + 6/((x+1)(x+5)`

our LCD is `(x-5)(x+5)(x+1)`

`4/((x-5)(x+5)) * ((x+1))/((x+1)) = (4(x+1))/((x-5)(x+5)(x+1)` and

`6/((x+1)(x+5)) * ((x-5))/((x-5)) = (6(x-5))/((x-5)(x+5)(x+1))`

`(4(x+1) + 6(x-5))/((x-5)(x+5)(x+1) )`

Distributive Property

`(4x + 4 + 6x - 30)/((x-5)(x+5)(x+1) )`

add like terms

`(10x - 26)/((x-5)(x+5)(x+1) )` Fully simplified