Question

Subtract the rational expressions x2/2x−12−18/x−6 . Simplify the answer.(1 point) Responses x+62 Start Fraction x plus 6 over 2 End Fraction x2−362(x−6) Start Fraction x squared minus 36 over 2 left parenthesis x minus 6 right parenthesis End Fraction x2−18x−6 Start Fraction x squared minus 18 over x minus 6 End Fraction x−62

Answer 1

To subtract the rational expressions \( \frac{x^2}{2x - 12} - \frac{18}{x - 6} \), we first need a common denominator.

1. **Factoring the denominators:**
- The first denominator can be factored:
\[
2x - 12 = 2(x - 6)
\]
- The second denominator \( x - 6 \) stays the same.

2. **Finding the common denominator:**
The common denominator of \( 2(x - 6) \) and \( x - 6 \) is \( 2(x - 6) \).

3. **Rewriting the fractions with a common denominator:**
\[
\frac{x^2}{2(x - 6)} - \frac{18}{x - 6} = \frac{x^2}{2(x - 6)} - \frac{18 \cdot 2}{2(x - 6)} = \frac{x^2 - 36}{2(x - 6)}
\]

(Here, we multiplied \( 18 \) by \( 2 \) to get the common denominator.)

4. **Simplifying \( x^2 - 36 \):**
The expression \( x^2 - 36 \) is a difference of squares, which factors as:
\[
x^2 - 36 = (x - 6)(x + 6)
\]

5. **Substituting back into the expression:**
Now we can rewrite the rational expression:
\[
\frac{(x - 6)(x + 6)}{2(x - 6)}
\]

6. **Canceling the common terms:**
Since \( x - 6 \) appears in both the numerator and the denominator, we can cancel it, assuming \( x \neq 6 \):
\[
= \frac{x + 6}{2}
\]

7. **Final simplified expression:**
Thus, the final simplified answer is:
\[
\frac{x + 6}{2}
\]

So the correct option is:
\[
\text{Start Fraction x plus 6 over 2 End Fraction}
\]