Add the rational expressions to find the sum: −4x−4+x2−2x2x−8 . Put the answer into its simplest form.(1 point)

Question

Responses

x2−2x−4x−4
Start Fraction x squared minus 2 x minus 4 over x minus 4 End Fraction

x+22
Start Fraction x plus 2 over 2 End Fraction

x2−2x−82x−8
Start Fraction x squared minus 2 x minus 8 over 2 x minus 8 End Fraction

x2−2x2x

Answer 1

To find the sum of the rational expressions \(-\frac{4}{x-4} + \frac{x^2 - 2}{2x - 8}\), we first need to rewrite the second expression.

Notice that \(2x - 8\) can be factored:

\[ 2x - 8 = 2(x - 4) \]

So, we can rewrite the second term:

\[ \frac{x^2 - 2}{2x - 8} = \frac{x^2 - 2}{2(x - 4)} = \frac{1}{2} \cdot \frac{x^2 - 2}{x - 4} \]

Now we can express the first term with a common denominator:

\[ -\frac{4}{x-4} = -\frac{4}{1} \cdot \frac{2}{2} \cdot \frac{1}{x-4} = -\frac{8}{2(x-4)} \]

Now we have:

\[ -\frac{8}{2(x - 4)} + \frac{x^2 - 2}{2(x - 4)} = \frac{-8 + (x^2 - 2)}{2(x - 4)} \]

Combine the numerators:

\[ = \frac{x^2 - 10}{2(x - 4)} \]

Thus, the sum of the rational expressions is:

\[ \frac{x^2 - 10}{2(x - 4)} \]

Now that we have added the rational expressions and simplified them, we can conclude that the final answer is:

\[ \frac{x^2 - 10}{2(x - 4)} \]

In simplest form, the answer is:

\[ \boxed{\frac{x^2 - 10}{2(x - 4)}} \]