To simplify the expression [tex]\(\frac{4x + 1}{x^2 - 4} - \frac{3}{x - 2}\)[/tex], let's go through the steps systematically.
1. Identify the denominators and factor:
We need a common denominator to subtract these fractions. Notice that:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
So, the first fraction is already over the denominator [tex]\((x - 2)(x + 2)\)[/tex], while the second fraction is over [tex]\((x - 2)\)[/tex].
2. Rewrite the second fraction to have the common denominator:
To subtract these fractions, the second fraction needs to be rewritten with the common denominator [tex]\((x - 2)(x + 2)\)[/tex]:
[tex]\[
\frac{3}{x - 2} = \frac{3 \cdot (x + 2)}{(x - 2)(x + 2)} = \frac{3(x + 2)}{(x - 2)(x + 2)}
\][/tex]
3. Subtract the fractions:
Now that both fractions have a common denominator, we can rewrite the overall expression as a single fraction:
[tex]\[
\frac{4x + 1}{(x - 2)(x + 2)} - \frac{3(x + 2)}{(x - 2)(x + 2)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{(4x + 1) - 3(x + 2)}{(x - 2)(x + 2)}
\][/tex]
4. Simplify the numerator:
Distribute [tex]\(3\)[/tex] in the second term and then combine like terms:
[tex]\[
(4x + 1) - 3(x + 2) = 4x + 1 - 3x - 6
\][/tex]
Combine the [tex]\(x\)[/tex] terms and the constants:
[tex]\[
4x - 3x = x
\][/tex]
[tex]\[
1 - 6 = -5
\][/tex]
Therefore, the simplified numerator is:
[tex]\[
x - 5
\][/tex]
5. Combine the results:
Putting it all together, we have:
[tex]\[
\frac{x - 5}{(x - 2)(x + 2)}
\][/tex]
Since [tex]\(x^2 - 4\)[/tex] is equivalent to [tex]\((x - 2)(x + 2)\)[/tex], we can write the final simplified fraction as:
[tex]\[
\frac{x - 5}{x^2 - 4}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\frac{x - 5}{x^2 - 4}
\][/tex]
