Certainly! Let's go through a step-by-step solution to subtract the fractions:
1. Identify and Factor the Denominators:
- The first fraction's denominator is [tex]\( x^2 - 36 \)[/tex].
This can be factored as a difference of squares:
[tex]\[
x^2 - 36 = (x - 6)(x + 6)
\][/tex]
- The second fraction's denominator is [tex]\( x^2 + 12x + 36 \)[/tex].
This can be factored as a perfect square trinomial:
[tex]\[
x^2 + 12x + 36 = (x + 6)^2
\][/tex]
So, our fractions now look like:
[tex]\[
\frac{4}{(x - 6)(x + 6)} - \frac{2}{(x + 6)^2}
\][/tex]
2. Find a Common Denominator:
- The common denominator for these fractions will be the Least Common Denominator (LCD):
[tex]\[
LCD = (x - 6)(x + 6)^2
\][/tex]
3. Rewrite Each Fraction with the Common Denominator:
- Rewrite the first fraction [tex]\(\frac{4}{(x - 6)(x + 6)}\)[/tex] with the common denominator:
Multiply the numerator and denominator by [tex]\((x + 6)\)[/tex] to get:
[tex]\[
\frac{4(x + 6)}{(x - 6)(x + 6)^2}
\][/tex]
- The second fraction [tex]\(\frac{2}{(x + 6)^2}\)[/tex] already needs the denominator [tex]\((x - 6)(x + 6)^2\)[/tex]:
Multiply the numerator and denominator by [tex]\((x - 6)\)[/tex] to get:
[tex]\[
\frac{2(x - 6)}{(x - 6)(x + 6)^2}
\][/tex]
4. Subtract the Fractions:
- Now that both fractions have the same denominator, we can subtract the numerators:
[tex]\[
\frac{4(x + 6) - 2(x - 6)}{(x - 6)(x + 6)^2}
\][/tex]
- Simplify the numerator:
[tex]\[
4(x + 6) = 4x + 24
\][/tex]
[tex]\[
2(x - 6) = 2x - 12
\][/tex]
[tex]\[
4x + 24 - (2x - 12) = 4x + 24 - 2x + 12 = 2x + 36
\][/tex]
So, the fraction now looks like:
[tex]\[
\frac{2x + 36}{(x - 6)(x + 6)^2}
\][/tex]
5. Factor the Numerator and Simplify:
- The numerator [tex]\(2x + 36\)[/tex] can be factored as:
[tex]\[
2(x + 18)
\][/tex]
- This gives us:
[tex]\[
\frac{2(x + 18)}{(x - 6)(x + 6)^2}
\][/tex]
- Simplify as far as possible.
Therefore, the simplified form of the given subtraction problem is:
[tex]\[
\frac{2(x + 18)}{(x - 6)(x + 6)^2}
\][/tex]
This is the final simplified expression.
