Question

progress:
the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your
perform the indicated operations, then simplify.
\\(\frac{6x^2}{x^2 - 36} - \frac{6x}{x + 6} + \frac{6}{x - 6}\\)
\\(\frac{6x^2 - 6x + 6}{(x - 6)(x + 6)}\\) \\(\frac{-30x + 36}{(x + 6)(x - 6)}\\) \\(\frac{42x + 36}{(x + 6)(x - 6)}\\) \\(\frac{42}{x}\\)

Explanation:

Step1: Factor the denominator

Notice that \(x^2 - 36\) is a difference of squares, so \(x^2 - 36=(x + 6)(x - 6)\). The common denominator for the three fractions is \((x + 6)(x - 6)\).

Step2: Rewrite each fraction with the common denominator

• For \(\frac{6x^2}{x^2 - 36}\), it remains \(\frac{6x^2}{(x + 6)(x - 6)}\) since the denominator is already factored as \((x + 6)(x - 6)\).
• For \(\frac{6x}{x + 6}\), multiply the numerator and denominator by \((x - 6)\) to get \(\frac{6x(x - 6)}{(x + 6)(x - 6)}=\frac{6x^2-36x}{(x + 6)(x - 6)}\).
• For \(\frac{6}{x - 6}\), multiply the numerator and denominator by \((x + 6)\) to get \(\frac{6(x + 6)}{(x + 6)(x - 6)}=\frac{6x + 36}{(x + 6)(x - 6)}\).

Step3: Perform the operations

Now, substitute these back into the original expression:

$$
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$$

Answer:

\(\boldsymbol{\frac{42x + 36}{(x + 6)(x - 6)}}\) (corresponding to the third option)