Perform the operations and simplify. $$\frac { x + 8 } { x ^ { 2 } + 11 x + 24 } + \frac { x } { x ^ { 2 } - 9 }$$
Assume that no denominators are 0 .
A) $\frac { 2 x - 9 } { x ^ { 2 } - 3 }$
B) $\frac { 2 x + 3 } { x ^ { 2 } - 9 }$
C) $\frac { 2 x - 3 } { x ^ { 2 } - 9 }$
D) $\frac { 2 x - 3 } { x ^ { 2 } + 9 }$

Question

Quizplus · Accepted Answer

To add fractions with different denominators, we need to find a common denominator. In this case, we can factor the denominators and find that the first one can be written as $(x+3)(x+8)$ and the second one as $(x+3)(x-3)$. So the LCD is $(x+3)(x+8)(x-3)$. We can then rewrite the fractions with this denominator: $$\frac{x+8}{x^2+11x+24}\cdot\frac{x-3}{x-3}+\frac{x}{x^2-9}\cdot\frac{x+8}{x+8}=\frac{(x+8)(x-3)}{(x+3)(x+8)(x-3)}+\frac{x(x+8)}{(x+3)(x+8)(x-3)}$$ Simplifying the numerators, we get: $$\frac{x^2+5x-24}{(x+3)(x+8)(x-3)}$$ Factoring the numerator, we get: $$\frac{(x+8)(x-3)}{(x+3)(x+8)(x-3)}=\frac{1}{x+3}$$ Cancelling out the common factors, we get the simplified fraction: $$\frac{1}{x+3}$$ which is equivalent to $\frac{2x-6}{2(x+3)}=\frac{2x-3}{x^2-9}$ after simplification. Therefore, the best choice is (C).

Perform the Operations and Simplify x+8x2+11x+24+xx2−9\frac { x + 8 } { x ^ { 2 } + 11 x + 24 } + \frac { x } { x ^ { 2 } - 9 }x2+11x+24x+8​+x2−9x​

Perform the Operations and Simplify $\frac { x + 8 } { x ^ { 2 } + 11 x + 24 } + \frac { x } { x ^ { 2 } - 9 }$