Answer :
To solve the problem of finding the difference between the two given fractions and expressing the answer in simplest form, let's proceed with the following steps:
1. Identify the Expressions:
We're working with the fractions:
[tex]\[
\frac{2s}{s^2 - 4s + 4} - \frac{4}{s^2 - 4s + 4}
\][/tex]
2. Find a Common Denominator:
Since both fractions already have the same denominator [tex]\(s^2 - 4s + 4\)[/tex], we can directly subtract the numerators.
3. Subtract the Numerators:
The difference in the numerators will be:
[tex]\[
(2s - 4)
\][/tex]
4. Write the Expression:
This gives us the expression:
[tex]\[
\frac{2s - 4}{s^2 - 4s + 4}
\][/tex]
5. Simplify the Expression:
Notice that both the numerator and the denominator can be factorized:
- The numerator, [tex]\(2s - 4\)[/tex], can be factored as [tex]\(2(s - 2)\)[/tex].
- The denominator, [tex]\(s^2 - 4s + 4\)[/tex], is a perfect square and can be factored as [tex]\((s - 2)^2\)[/tex].
So the expression becomes:
[tex]\[
\frac{2(s - 2)}{(s - 2)^2}
\][/tex]
6. Cancel Common Factors:
We can cancel one factor of [tex]\((s - 2)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{2}{s - 2}
\][/tex]
Therefore, the difference expressed in its simplest form is:
[tex]\[
\frac{2}{s - 2}
\][/tex]
1. Identify the Expressions:
We're working with the fractions:
[tex]\[
\frac{2s}{s^2 - 4s + 4} - \frac{4}{s^2 - 4s + 4}
\][/tex]
2. Find a Common Denominator:
Since both fractions already have the same denominator [tex]\(s^2 - 4s + 4\)[/tex], we can directly subtract the numerators.
3. Subtract the Numerators:
The difference in the numerators will be:
[tex]\[
(2s - 4)
\][/tex]
4. Write the Expression:
This gives us the expression:
[tex]\[
\frac{2s - 4}{s^2 - 4s + 4}
\][/tex]
5. Simplify the Expression:
Notice that both the numerator and the denominator can be factorized:
- The numerator, [tex]\(2s - 4\)[/tex], can be factored as [tex]\(2(s - 2)\)[/tex].
- The denominator, [tex]\(s^2 - 4s + 4\)[/tex], is a perfect square and can be factored as [tex]\((s - 2)^2\)[/tex].
So the expression becomes:
[tex]\[
\frac{2(s - 2)}{(s - 2)^2}
\][/tex]
6. Cancel Common Factors:
We can cancel one factor of [tex]\((s - 2)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{2}{s - 2}
\][/tex]
Therefore, the difference expressed in its simplest form is:
[tex]\[
\frac{2}{s - 2}
\][/tex]
