`x/(x^2-9) + (x+1)/(x^2+6x+9)` Perform the indicated operation(s) and simplify

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gsarora17 eNotes educator| Certified Educator

`x/(x^2-9)+(x+1)/(x^2+6x+9)`

Apply the following identities to factorize the denominators of the above rational functions:

`a^2-b^2=(a+b)(a-b)`  and

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

`x/(x^2-9)+(x+1)/(x^2+6x+9)=x/(x^2-3^2)+(x+1)/(x^2+2x(3)+3^2)`

`=x/((x+3)(x-3))+(x+1)/(x+3)^2`

LCD of the above expression is `(x-3)(x+3)^2`

`=(x(x+3)+(x+1)(x-3))/((x-3)(x+3)^2)` 

`=(x^2+3x+x^2-3x+x-3)/((x-3)(x+3)^2)`

Combine the like terms of the numerator,

`=(x^2+x^2+3x-3x+x-3)/((x-3)(x+3)^2)`

`=(2x^2+x-3)/((x-3)(x+3)^2)`

Factorize the numerator by splitting the middle term,

`=(2x^2-2x+3x-3)/((x-3)(x+3)^2)`

`=(2x(x-1)+3(x-1))/((x-3)(x+3)^2)`

`=((2x+3)(x-1))/((x-3)(x+3)^2)`