Question
Add the rational expressions to find the sum:
x−4
−4
+
2x−8
x
2
−2x
Put the answer into its simplest form. A.
2
x+2
B.
2x
x
4
−2x
C.
2x−8
x
2
−2x−8
D.
x−4
x
2
−4
x−4
−4
+
2x−8
x
2
−2x
Put the answer into its simplest form. A.
2
x+2
B.
2x
x
4
−2x
C.
2x−8
x
2
−2x−8
D.
x−4
x
2
−4
Answers
GPT-4o mini
To find the sum of the rational expressions \(\frac{x - 4}{-4} + \frac{2x - 8}{x^2 - 2x}\), we first need to simplify the second expression \(\frac{2x - 8}{x^2 - 2x}\).
1. **Factor the second expression**:
- The numerator \(2x - 8\) can be factored to \(2(x - 4)\).
- The denominator \(x^2 - 2x\) can be factored to \(x(x - 2)\).
Thus, we can rewrite the second fraction:
\[
\frac{2x - 8}{x^2 - 2x} = \frac{2(x - 4)}{x(x - 2)}.
\]
Now, our sum becomes:
\[
\frac{x - 4}{-4} + \frac{2(x - 4)}{x(x - 2)}.
\]
2. **Find a common denominator**:
The common denominator for these two fractions is \(-4x(x - 2)\).
3. **Rewrite each fraction with the common denominator**:
- The first fraction becomes:
\[
\frac{x - 4}{-4} \cdot \frac{x(x - 2)}{x(x - 2)} = \frac{(x - 4)x(x - 2)}{-4} = \frac{-(x - 4)x(x - 2)}{4}.
\]
- The second fraction becomes:
\[
\frac{2(x - 4)}{x(x - 2)} \cdot \frac{-4}{-4} = \frac{-8(x - 4)}{4x(x - 2)}.
\]
Now we can add the two fractions:
\[
\frac{-(x - 4)x(x - 2) - 8(x - 4)}{4(-x(x - 2))}.
\]
4. **Combine the numerators**:
Factor out \(-(x - 4)\) from the numerator:
\[
-(x - 4) \left(x(x - 2) + 8\right) = -(x - 4)(x^2 - 2x + 8).
\]
So our expression now becomes:
\[
\frac{-(x - 4)(x^2 - 2x + 8)}{4(-x(x - 2))} = \frac{(x - 4)(x^2 - 2x + 8)}{4x(x - 2)}.
\]
5. **Final simplification**:
We have the final expression as:
\[
\frac{(x - 4)(x^2 - 2x + 8)}{4x(x - 2)}.
\]
6. **Choose the correct option**:
Now, we can compare this expression with the options provided:
- A. \( \frac{2}{x + 2} \)
- B. \( \frac{2x}{x^4 - 2x} \)
- C. \( \frac{2x - 8}{x^2 - 2x - 8} \)
- D. \( \frac{x - 4}{x^2 - 4} \)
None of the options match our derived expression. It's possible that there may have been a mistake in the options or the problem setup.
However, the simplified sum of the given rational expressions is:
\[
\frac{(x - 4)(x^2 - 2x + 8)}{4x(x - 2)}.
\]
If you have further clarifications or need different details, feel free to ask!
1. **Factor the second expression**:
- The numerator \(2x - 8\) can be factored to \(2(x - 4)\).
- The denominator \(x^2 - 2x\) can be factored to \(x(x - 2)\).
Thus, we can rewrite the second fraction:
\[
\frac{2x - 8}{x^2 - 2x} = \frac{2(x - 4)}{x(x - 2)}.
\]
Now, our sum becomes:
\[
\frac{x - 4}{-4} + \frac{2(x - 4)}{x(x - 2)}.
\]
2. **Find a common denominator**:
The common denominator for these two fractions is \(-4x(x - 2)\).
3. **Rewrite each fraction with the common denominator**:
- The first fraction becomes:
\[
\frac{x - 4}{-4} \cdot \frac{x(x - 2)}{x(x - 2)} = \frac{(x - 4)x(x - 2)}{-4} = \frac{-(x - 4)x(x - 2)}{4}.
\]
- The second fraction becomes:
\[
\frac{2(x - 4)}{x(x - 2)} \cdot \frac{-4}{-4} = \frac{-8(x - 4)}{4x(x - 2)}.
\]
Now we can add the two fractions:
\[
\frac{-(x - 4)x(x - 2) - 8(x - 4)}{4(-x(x - 2))}.
\]
4. **Combine the numerators**:
Factor out \(-(x - 4)\) from the numerator:
\[
-(x - 4) \left(x(x - 2) + 8\right) = -(x - 4)(x^2 - 2x + 8).
\]
So our expression now becomes:
\[
\frac{-(x - 4)(x^2 - 2x + 8)}{4(-x(x - 2))} = \frac{(x - 4)(x^2 - 2x + 8)}{4x(x - 2)}.
\]
5. **Final simplification**:
We have the final expression as:
\[
\frac{(x - 4)(x^2 - 2x + 8)}{4x(x - 2)}.
\]
6. **Choose the correct option**:
Now, we can compare this expression with the options provided:
- A. \( \frac{2}{x + 2} \)
- B. \( \frac{2x}{x^4 - 2x} \)
- C. \( \frac{2x - 8}{x^2 - 2x - 8} \)
- D. \( \frac{x - 4}{x^2 - 4} \)
None of the options match our derived expression. It's possible that there may have been a mistake in the options or the problem setup.
However, the simplified sum of the given rational expressions is:
\[
\frac{(x - 4)(x^2 - 2x + 8)}{4x(x - 2)}.
\]
If you have further clarifications or need different details, feel free to ask!