Question

add as indicated.
\\(\frac{2x}{x + 16} + \frac{x + 16}{x - 16}\\)

select the correct choice below and fill in the answer box(es) to complete your choice.
(simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)

\\(\bigcirc\\) a. \\(\frac{2x}{x + 16} + \frac{x + 16}{x - 16} = \square\\), no numbers must be excluded.

\\(\bigcirc\\) b. \\(\frac{2x}{x + 16} + \frac{x + 16}{x - 16} = \square\\), \\(x \
eq \square\\)

Explanation:

Step1: Find a common denominator

The denominators are \(x + 16\) and \(x - 16\), so the common denominator is \((x + 16)(x - 16)\).
Rewrite each fraction with the common denominator:
\(\frac{2x}{x + 16} = \frac{2x(x - 16)}{(x + 16)(x - 16)}\)
\(\frac{x + 16}{x - 16} = \frac{(x + 16)(x + 16)}{(x + 16)(x - 16)}\)

Step2: Add the fractions

\(\frac{2x(x - 16) + (x + 16)^2}{(x + 16)(x - 16)}\)
Expand the numerators:
\(2x(x - 16) = 2x^2 - 32x\)
\((x + 16)^2 = x^2 + 32x + 256\)
Add them together: \(2x^2 - 32x + x^2 + 32x + 256 = 3x^2 + 256\)
So the fraction becomes \(\frac{3x^2 + 256}{(x + 16)(x - 16)}\) or \(\frac{3x^2 + 256}{x^2 - 256}\)

Step3: Determine excluded values

The denominators \(x + 16\) and \(x - 16\) cannot be zero, so \(x
eq -16\) and \(x
eq 16\). But looking at the options, option B has one excluded value box, but actually there are two. However, maybe the problem expects us to note that \(x\) cannot be -16 or 16, but since option B has one box, maybe a typo, but we proceed. The simplified expression is \(\frac{3x^2 + 256}{x^2 - 256}\) and \(x
eq -16, 16\), but as per the option B format, we can put one (but actually two). But let's check the options again. Option B says \(x
eq \square\), maybe the problem expects us to list one, but actually both -16 and 16 are excluded. But maybe in the problem's context, we can proceed with the simplified expression and note the excluded values.

Answer:

B. \(\frac{2x}{x + 16} + \frac{x + 16}{x - 16} = \frac{3x^2 + 256}{x^2 - 256}\), \(x
eq -16, 16\) (but as per the option's single box, maybe a mistake, but the simplified expression is \(\frac{3x^2 + 256}{x^2 - 256}\) and excluded values are -16 and 16)