Question

`(x - 2)/(x + 3) + \frac{10x}{x {}^(2 ) - 9}`simplify the sum. state any restrictions on the variables.

Answer 1

We have

`(x-2)/(x+3)+\frac{10x}{x{}^2-9}`

first, we need to factorize the next term

`x^2-9=(x+3)(x-3)`

so we have

`(x-2)/(x+3)+(10x)/((x+3)(x-3))`

Remember in order to sum a fraction the denominator must be the same

`((x-2)(x-3)+10x)/((x+3)(x-3))`

then we solve the multiplications (x-2)(x-3)

`(x^2-3x-2x+6+10x)/((x+3)(x-3))=(x^2+5x+6)/((x+3)(x-3))`

then we can factorize the numerator

`x^2+5x+6=(x+3)(x+2)`

so the simplification will be

`(x^2+5x+6)/((x+3)(x-3))=((x+3)(x+2))/((x+3)(x-3))=((x+2))/((x-3))`

the final result is

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