Answer :
Let's find the difference between the two fractions and express the answer in its simplest form.
We have two fractions with the same denominator:
[tex]\(\frac{6s}{s^2 - 4s + 4}\)[/tex] and [tex]\(\frac{12}{s^2 - 4s + 4}\)[/tex].
### Step 1: Subtract the Numerators
Since the denominators are the same, we can subtract the numerators directly:
[tex]\[
\frac{6s}{s^2 - 4s + 4} - \frac{12}{s^2 - 4s + 4} = \frac{6s - 12}{s^2 - 4s + 4}.
\][/tex]
### Step 2: Simplify the Numerator
The numerator [tex]\(6s - 12\)[/tex] can be factored by taking out the greatest common factor, which is 6:
[tex]\[
6s - 12 = 6(s - 2).
\][/tex]
### Step 3: Simplify the Entire Expression
Now we have:
[tex]\[
\frac{6(s - 2)}{s^2 - 4s + 4}.
\][/tex]
We need to check if the denominator [tex]\(s^2 - 4s + 4\)[/tex] can be simplified or factored. It can be factored as:
[tex]\[
s^2 - 4s + 4 = (s - 2)^2.
\][/tex]
### Step 4: Cancel Out Common Factors
Notice we now have:
[tex]\[
\frac{6(s - 2)}{(s - 2)^2}.
\][/tex]
We can cancel one [tex]\((s - 2)\)[/tex] term from the numerator with one [tex]\((s - 2)\)[/tex] term from the denominator, leaving us with:
[tex]\[
\frac{6}{s - 2}.
\][/tex]
So, the simplified form of the difference is:
[tex]\[
\frac{6}{s - 2}.
\][/tex]
This is the simplest form of the expression.
We have two fractions with the same denominator:
[tex]\(\frac{6s}{s^2 - 4s + 4}\)[/tex] and [tex]\(\frac{12}{s^2 - 4s + 4}\)[/tex].
### Step 1: Subtract the Numerators
Since the denominators are the same, we can subtract the numerators directly:
[tex]\[
\frac{6s}{s^2 - 4s + 4} - \frac{12}{s^2 - 4s + 4} = \frac{6s - 12}{s^2 - 4s + 4}.
\][/tex]
### Step 2: Simplify the Numerator
The numerator [tex]\(6s - 12\)[/tex] can be factored by taking out the greatest common factor, which is 6:
[tex]\[
6s - 12 = 6(s - 2).
\][/tex]
### Step 3: Simplify the Entire Expression
Now we have:
[tex]\[
\frac{6(s - 2)}{s^2 - 4s + 4}.
\][/tex]
We need to check if the denominator [tex]\(s^2 - 4s + 4\)[/tex] can be simplified or factored. It can be factored as:
[tex]\[
s^2 - 4s + 4 = (s - 2)^2.
\][/tex]
### Step 4: Cancel Out Common Factors
Notice we now have:
[tex]\[
\frac{6(s - 2)}{(s - 2)^2}.
\][/tex]
We can cancel one [tex]\((s - 2)\)[/tex] term from the numerator with one [tex]\((s - 2)\)[/tex] term from the denominator, leaving us with:
[tex]\[
\frac{6}{s - 2}.
\][/tex]
So, the simplified form of the difference is:
[tex]\[
\frac{6}{s - 2}.
\][/tex]
This is the simplest form of the expression.
