To convert the fraction [tex]\(\frac{1}{w+5}\)[/tex] into an equivalent fraction with the denominator [tex]\(w^2 + w - 20\)[/tex], follow these steps:
1. Factor the target denominator [tex]\(w^2 + w - 20\)[/tex]:
[tex]\[w^2 + w - 20\][/tex]
To factor this quadratic expression, we look for two numbers that multiply to [tex]\(-20\)[/tex] and add to [tex]\(1\)[/tex], specifically [tex]\((4)\)[/tex] and [tex]\((-5)\)[/tex]. Thus, we can write:
[tex]\[w^2 + w - 20 = (w + 5)(w - 4)\][/tex]
2. Express the original fraction with the new denominator:
We start with the fraction [tex]\(\frac{1}{w+5}\)[/tex] and want to rewrite it with the denominator [tex]\(w^2 + w - 20\)[/tex]. We know that:
[tex]\[w^2 + w - 20 = (w + 5)(w - 4)\][/tex]
We can multiply both the numerator and the denominator of [tex]\(\frac{1}{w+5}\)[/tex] by [tex]\((w - 4)\)[/tex] to achieve the new denominator:
[tex]\[
\frac{1}{w+5} \times \frac{w-4}{w-4} = \frac{w - 4}{(w+5)(w-4)}
\][/tex]
3. Simplify the new equivalent fraction:
The denominator is now clearly [tex]\(w^2 + w - 20\)[/tex], so the fraction becomes:
[tex]\[
\frac{w - 4}{w^2 + w - 20}
\][/tex]
4. Compare with the answer choices:
Based on the answer choices given:
- Option A: [tex]\(\frac{w^2-7w-12}{w^2+w-20}\)[/tex]
- Option B: [tex]\(\frac{w^2+7w+12}{w^2+w-20}\)[/tex]
- Option C: [tex]\(\frac{w^2+7w-12}{w^2+w-20}\)[/tex]
- Option D: [tex]\(\frac{w^2-7w+12}{w^2+w-20}\)[/tex]
None of these numerators exactly matches [tex]\(w - 4\)[/tex], but we see that the numerator we have in [tex]\(\frac{w - 4}{w^2 + w - 20}\)[/tex] is indeed in its simplest form as [tex]\(w\)[/tex].
Thus, none of the above options are in direct correspondence with our simplified fraction. However, it's likely a simplification step or transformation may have been missed in framing the multiple choice options.
Nevertheless, based on what we derived:
The equivalent fraction is:
Answer: [tex]\(\frac{w - 4}{w^2 + w - 20}\)[/tex]
But since this precise form does not appear as an option, none of the options is correct based on the exact transformation.
