Question

cw: solving rational equations)
lesson 2.5 review - solving rational equations
② \\(\frac{2a^2}{a^2 - 25} - \frac{a}{(a + 5)} = \frac{7}{(a - 5)}\\)
\\(\frac{2a^2}{(a + 5)(a - 5)} - \frac{a}{(a + 5)} \underline{\quad\quad} = \frac{7}{(a - 5)} \frac{(a + 5)}{(a + 5)}\\)
lcd = (a + 5)
① factor denominators
② find lcd
③ multiply to create lcd
④ solve using numerator
⑤ check for extraneous solutions!
multiple - choice question
what does the \\(-\frac{a}{(a + 5)}\\) need to have in its denominator in order to have a least common denominator with the other 2 fractions?
\\((a - 5)\\)
\\((a + 5)\\)
rewatch
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Explanation:

Step1: Analyze denominators

The other two fractions have denominators \((a + 5)(a - 5)\) (from \(\frac{2a^2}{a^2 - 25}\) since \(a^2 - 25=(a + 5)(a - 5)\)) and \((a - 5)\) (from \(\frac{7}{a - 5}\)). The fraction \(-\frac{a}{a + 5}\) currently has denominator \((a + 5)\).

Step2: Determine needed factor

To get a least common denominator (LCD) with the other two fractions, we need to include the factor that the current denominator of \(-\frac{a}{a + 5}\) is missing. The LCD of \((a + 5)(a - 5)\), \((a - 5)\), and \((a + 5)\) is \((a + 5)(a - 5)\). Since \(-\frac{a}{a + 5}\) has \((a + 5)\) in the denominator, it needs \((a - 5)\) to match the LCD.

Answer:

\((a - 5)\)