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Answer :
To find the difference between the two fractions and express it in its simplest form, we need to follow these steps:
1. Identify the fractions:
- The first fraction is [tex]\(\frac{3s}{s^2 - 4s + 4}\)[/tex].
- The second fraction is [tex]\(\frac{6}{s^2 - 4s + 4}\)[/tex].
2. Note the common denominator:
- Both fractions have the same denominator: [tex]\(s^2 - 4s + 4\)[/tex].
3. Subtract the fractions:
- Since the denominators are the same, subtract the numerators directly:
[tex]\[
\frac{3s}{s^2 - 4s + 4} - \frac{6}{s^2 - 4s + 4} = \frac{3s - 6}{s^2 - 4s + 4}
\][/tex]
4. Simplify the numerator:
- The expression [tex]\(3s - 6\)[/tex] can be factored:
[tex]\[
3s - 6 = 3(s - 2)
\][/tex]
5. Rewrite the expression:
- Substitute the factored form into the expression:
[tex]\[
\frac{3(s - 2)}{s^2 - 4s + 4}
\][/tex]
6. Factor the denominator:
- Notice that [tex]\(s^2 - 4s + 4\)[/tex] can be factored as [tex]\((s - 2)^2\)[/tex], based on the quadratic expression:
[tex]\[
s^2 - 4s + 4 = (s - 2)(s - 2)
\][/tex]
7. Cancel common terms:
- Because we have [tex]\((s - 2)\)[/tex] in both the numerator and denominator, cancel one [tex]\((s - 2)\)[/tex] from each:
[tex]\[
\frac{3(s - 2)}{(s - 2)(s - 2)} = \frac{3}{s - 2} \quad \text{for} \quad s \neq 2
\][/tex]
So, the simplified difference is [tex]\(\frac{3}{s - 2}\)[/tex]. This expresses the original subtraction in its simplest form, as long as [tex]\(s \neq 2\)[/tex] (to avoid division by zero).
1. Identify the fractions:
- The first fraction is [tex]\(\frac{3s}{s^2 - 4s + 4}\)[/tex].
- The second fraction is [tex]\(\frac{6}{s^2 - 4s + 4}\)[/tex].
2. Note the common denominator:
- Both fractions have the same denominator: [tex]\(s^2 - 4s + 4\)[/tex].
3. Subtract the fractions:
- Since the denominators are the same, subtract the numerators directly:
[tex]\[
\frac{3s}{s^2 - 4s + 4} - \frac{6}{s^2 - 4s + 4} = \frac{3s - 6}{s^2 - 4s + 4}
\][/tex]
4. Simplify the numerator:
- The expression [tex]\(3s - 6\)[/tex] can be factored:
[tex]\[
3s - 6 = 3(s - 2)
\][/tex]
5. Rewrite the expression:
- Substitute the factored form into the expression:
[tex]\[
\frac{3(s - 2)}{s^2 - 4s + 4}
\][/tex]
6. Factor the denominator:
- Notice that [tex]\(s^2 - 4s + 4\)[/tex] can be factored as [tex]\((s - 2)^2\)[/tex], based on the quadratic expression:
[tex]\[
s^2 - 4s + 4 = (s - 2)(s - 2)
\][/tex]
7. Cancel common terms:
- Because we have [tex]\((s - 2)\)[/tex] in both the numerator and denominator, cancel one [tex]\((s - 2)\)[/tex] from each:
[tex]\[
\frac{3(s - 2)}{(s - 2)(s - 2)} = \frac{3}{s - 2} \quad \text{for} \quad s \neq 2
\][/tex]
So, the simplified difference is [tex]\(\frac{3}{s - 2}\)[/tex]. This expresses the original subtraction in its simplest form, as long as [tex]\(s \neq 2\)[/tex] (to avoid division by zero).
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