To find the correct option for the equation \( \frac{a}{a-5} - \frac{2a+5}{1} = 50a^2 - 25 \) rewritten using its least common denominator (LCD), we must first determine that LCD.
- The denominators are \( a - 5 \) and \( 1 \). The least common denominator is simply \( a - 5 \).
Now, we rewrite the equation using the LCD:
\[ \frac{a}{a-5} - \frac{2a+5}{1} = 50a^2 - 25 \]
To eliminate the denominators, multiply the whole equation by \( (a-5) \):
\[ a - (2a + 5)(a - 5) = (50a^2 - 25)(a - 5) \]
We can expand this:
On the left: \[ a - (2a^2 - 10a + 5a - 25) = a - (2a^2 - 5a + 25) = a - 2a^2 + 5a - 25 = -2a^2 + 6a - 25 \]
On the right: \[ (50a^2 - 25)(a - 5) = 50a^3 - 250a^2 - 25a + 125 \]
Thus, the rewritten equation should be in terms of both sides correctly matching the terms in their factored forms.
Now let's evaluate the options provided:
- Option #1: \( a(a+5)(a-5)(a+5) - 2(a+5)(a+5)(a+5) = 50a^2 - 25 \)
- Option #2: \( a(a+5)(a-5)(a+5) - 2(a+5)(a+5)(a+5) = 50(a+5)(a^2 - 25)(a+5) \)
- Option #3: \( a(a+5)(a-5)(a+5) - 2(a-5)(a+5)(a-5) = 50a^2 - 25 \)
- Option #4: \( a(a+5)(a-5)(a+5) - 2(a-5)(a+5)(a-5) = 50(a-5)(a^2 - 25)(a-5) \)
After carefully evaluating each option using the LCD: The correct version should utilize terms appropriately with their least common denominators.
Given these analyses and the options, it is appropriate to conclude the least common denominator usage aligns best with an identification of \( 50(a-5)(a^2 - 25)(a-5) \).
Thus, the answer is Option #4.
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