Question

add. enter your answer as a single, simplified, rational expression where the numerator and denominator have no common factors.\\(\frac{x + 8}{x - 8} + \frac{x - 6}{x + 6}\\)\
\\(4x/((x - 8)(x - 6))\\)\\(\frac{4x}{(x - 8)(x - 6)}\\)\
thats not it.\
thats not it. try again. score: 0%

Explanation:

Step1: Find a common denominator

The common denominator of \(\frac{x + 8}{x - 8}\) and \(\frac{x - 6}{x + 6}\) is \((x - 8)(x + 6)\).
Rewrite each fraction with the common denominator:
\(\frac{(x + 8)(x + 6)}{(x - 8)(x + 6)}+\frac{(x - 6)(x - 8)}{(x - 8)(x + 6)}\)

Step2: Expand the numerators

Expand \((x + 8)(x + 6)\):
\((x + 8)(x + 6)=x^2+6x + 8x+48=x^2 + 14x+48\)
Expand \((x - 6)(x - 8)\):
\((x - 6)(x - 8)=x^2-8x-6x + 48=x^2-14x + 48\)

Step3: Add the numerators

Add the expanded numerators:
\(\frac{x^2 + 14x+48+x^2-14x + 48}{(x - 8)(x + 6)}\)
Combine like terms in the numerator:
\(x^2+x^2+14x-14x + 48+48 = 2x^2+96\)

Step4: Simplify the numerator

Factor out a 2 from the numerator:
\(\frac{2(x^2 + 48)}{(x - 8)(x + 6)}\)
We can also expand the denominator: \((x - 8)(x + 6)=x^2-2x-48\)
So the simplified expression is \(\frac{2x^2 + 96}{x^2-2x - 48}\) or \(\frac{2(x^2 + 48)}{(x - 8)(x + 6)}\) (both are correct, and the numerator and denominator have no common factors).

Answer:

\(\frac{2x^2 + 96}{x^2-2x - 48}\) (or \(\frac{2(x^2 + 48)}{(x - 8)(x + 6)}\))