Question

How do you add or subtract `(2)/(x^2 + 8x + 15) + (1)/(x^2 + 11x + 30)`?

Answer 1

Answer is `3/((x+3)(x+6)` OR `3/(x^2+9x+18)`

Explanation:

First we simplify and find the factors of `x^2+8x+15` and `x^2+11x+30`.

Step 1: Factors of `x^2+8x+15` equation
`x^2+8x+15` -----> factors are 3 and 5 (`3+5=8` and `3xx5=15`)
`x^2+8x+15`
`x^2+3x+5x+15`
`x(x+3)+5(x+3)`
`(color(red)(x+3))(color(red)(x+5))`

Step 2: Factors of `x^2+11x+30`.
`x^2+11x+30` ----> factors are 5 and 6 (`5+6=11` and `5xx6=30)`
`x^2+11x+30`
`x^2+6x+5x+30`
`x(x+6)+5(x+6)`
`(color(red)(x+6))(color(red)(x+5))`

Step 3: Re-write the original equation with the above common factors and simplify further

`2/((x+3)(x+5))` + `1/((x+6)(x+5))`

Least Common Multiple is `(x+3)(x+6)(x+5)`

`(2(x+6)+1(x+3))/((x+3)(x+6)(x+5))`

`(2x+12+x+3)/((x+3)(x+6)(x+5))`

`(3x+15)/((x+3)(x+6)(x+5))`-----> simplify `3x+15`

`(3(x+5))/((x+3)(x+6)(x+5))`

`(3cancel((x+5)))/((x+3)(x+6)cancel(x+5))`

`3/((x+3)(x+6)` OR `3/(x^2+9x+18)`