Sure, let's simplify the given expression step-by-step:
The given expression is:
[tex]\[
\frac{a+2}{2 a+2}-\frac{7 a}{8 a^2-8}-\frac{a-3}{4 a-4}
\][/tex]
First, observe that we can factorize the denominators to potentially simplify the expression.
1. Simplify the first term:
[tex]\[
\frac{a+2}{2 a+2}
\][/tex]
Notice that [tex]\(2a + 2\)[/tex] can be factored as [tex]\(2(a + 1)\)[/tex], so:
[tex]\[
\frac{a+2}{2(a+1)} = \frac{a+2}{2(a+1)}
\][/tex]
2. Simplify the second term:
[tex]\[
\frac{7a}{8a^2-8}
\][/tex]
Notice that [tex]\(8a^2 - 8\)[/tex] can be factored as [tex]\(8(a^2 - 1)\)[/tex], which further factors to [tex]\(8(a-1)(a+1)\)[/tex]:
[tex]\[
\frac{7a}{8(a-1)(a+1)}
\][/tex]
3. Simplify the third term:
[tex]\[
\frac{a-3}{4a-4}
\][/tex]
Notice that [tex]\(4a - 4\)[/tex] can be factored as [tex]\(4(a-1)\)[/tex]:
[tex]\[
\frac{a-3}{4(a-1)}
\][/tex]
Now, the expression is:
[tex]\[
\frac{a+2}{2(a+1)} - \frac{7a}{8(a-1)(a+1)} - \frac{a-3}{4(a-1)}
\][/tex]
To combine these fractions, we need a common denominator. The common denominator will be [tex]\(8(a-1)(a+1)\)[/tex].
- Rewrite the first term with the common denominator:
[tex]\[
\frac{a+2}{2(a+1)} = \frac{(a+2) \cdot 4(a-1)}{8(a-1)(a+1)} = \frac{4(a+2)(a-1)}{8(a-1)(a+1)}
\][/tex]
- Rewrite the second term:
[tex]\[
\frac{7a}{8(a-1)(a+1)} \text{ (already in the common denominator)}
\][/tex]
- Rewrite the third term with the common denominator:
[tex]\[
\frac{a-3}{4(a-1)} = \frac{(a-3) \cdot 2(a+1)}{8(a-1)(a+1)} = \frac{2(a-3)(a+1)}{8(a-1)(a+1)}
\][/tex]
Now the expression becomes:
[tex]\[
\frac{4(a+2)(a-1) - 7a - 2(a-3)(a+1)}{8(a-1)(a+1)}
\][/tex]
Next, expand and simplify the numerator:
- Expand [tex]\(4(a+2)(a-1)\)[/tex]:
[tex]\[
4(a^2 - a + 2a - 2) = 4(a^2 + a - 2)
\][/tex]
- Expand and simplify [tex]\(2(a-3)(a+1)\)[/tex]:
[tex]\[
2(a^2 + a - 3a - 3) = 2(a^2 - 2a - 3)
\][/tex]
So the numerator becomes:
[tex]\[
4(a^2 + a - 2) - 7a - 2(a^2 - 2a - 3)
\][/tex]
Combine like terms:
[tex]\[
= 4a^2 + 4a - 8 - 7a - 2a^2 + 4a + 6
= (4a^2 - 2a^2) + (4a - 7a + 4a) + (-8 + 6)
= 2a^2 + a - 2
\][/tex]
Therefore, we have:
[tex]\[
\frac{2a^2 + a - 2}{8(a-1)(a+1)}
\][/tex]
This gives us the simplified expression:
[tex]\[
\frac{2a^2 + a - 2}{8(a^2 - 1)}
\][/tex]
