Question

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)} quad = \frac{7}{(a - 5)} quad)
① factor denominators
② find lcd
③ multiply to create lcd
④ solve using numerator
⑤ check for extraneous solutions!
multiple - choice question
what is the lcd that we will ultimately force all
the fractions to have?
((a + 5)(a - 5))
((a - 5)^2)
((a + 5)^2)
rewatch

Explanation:

Step1: Factor Denominators

First, factor the denominator \(a^2 - 25\). Using the difference of squares formula \(x^2 - y^2=(x + y)(x - y)\), we get \(a^2 - 25=(a + 5)(a - 5)\). So the denominators are \((a + 5)(a - 5)\), \((a + 5)\), and \((a - 5)\).

Step2: Find LCD

The Least Common Denominator (LCD) of a set of fractions is the least common multiple of their denominators. To find the LCD, we take the highest power of each distinct factor. The factors here are \((a + 5)\) and \((a - 5)\). The highest power of \((a + 5)\) is 1 (from \((a + 5)\)) and the highest power of \((a - 5)\) is 1 (from \((a - 5)\)). So the LCD is the product of these highest powers, which is \((a + 5)(a - 5)\).

Answer:

A. \((a + 5)(a - 5)\)