Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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]
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]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
