Simplify the complex fraction. $\frac { - \frac { 7 } { x } - \frac { 14 } { x + 7 } } { \frac { 2 } { x ^ { 2 } + 7 x } + \frac { 2 } { x } }$
A) $- \frac { x + 7 } { 2 ( x + 8 ) }$
B) $- \frac { 3 x - 7 } { 7 ( x + 8 ) }$
C) $- \frac { 7 ( 3 x - 7 ) } { 2 ( x + 8 ) }$
D) $- \frac { 2 ( 3 x - 7 ) } { 7 ( x + 8 ) }$
E) $- \frac { 7 ( 3 x - 7 ) } { 2 ( x + 8 ) }$.

Question

Quizplus · Accepted Answer

To simplify the complex fraction, we need to first find a common denominator for the numerator and denominator. The denominator already has a common denominator, which is x(x+7). The numerator can be rewritten as:

$$\frac{-7(x+7)-14x}{x(x+7)}$$

Simplifying, we get:

$$\frac{-21x-49}{x(x+7)}$$

Now, let's simplify the denominator of the original fraction:

$$\frac{2}{x(x+7)} + \frac{2(x+7)}{x(x+7)} = \frac{2+2x+14}{x(x+7)} = \frac{2x+16}{x(x+7)}$$

Finally, we can simplify the complex fraction:

$$\frac{-21x-49}{x(x+7)} \div \frac{2x+16}{x(x+7)} = \frac{-21x-49}{x(x+7)} \cdot \frac{x(x+7)}{2x+16}$$

Canceling out the common factors, we get:

$$\frac{-7(3x-7)}{2(x+8)}$$

Therefore, the correct choice is E.

Simplify the Complex Fraction −7x−14x+72x2+7x+2x\frac { - \frac { 7 } { x } - \frac { 14 } { x + 7 } } { \frac { 2 } { x ^ { 2 } + 7 x } + \frac { 2 } { x } }x2+7x2​+x2​−x7​−x+714​​

Simplify the Complex Fraction $\frac { - \frac { 7 } { x } - \frac { 14 } { x + 7 } } { \frac { 2 } { x ^ { 2 } + 7 x } + \frac { 2 } { x } }$