Question

Subtract the rational expressions x22x−12−18x−6 . Simplify the answer.(1 point)
Responses

x2−18x−6
Start Fraction x squared minus 18 over x minus 6 End Fraction

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

x−62

Answer 1

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

1. **Factor the denominators:**

The first denominator can be factored as:
\[
2x - 12 = 2(x - 6)
\]
So the expression becomes:
\[
\frac{x^2}{2(x - 6)} - \frac{18}{x - 6}
\]

2. **Find a common denominator:**

The common denominator will be \(2(x - 6)\).

3. **Rewrite the second fraction with the common denominator:**

To get the second fraction to have the common denominator, multiply the numerator and denominator by 2:
\[
\frac{18}{x - 6} = \frac{18 \cdot 2}{(x - 6) \cdot 2} = \frac{36}{2(x - 6)}
\]

4. **Now, subtract the fractions:**

Now we can subtract:
\[
\frac{x^2}{2(x - 6)} - \frac{36}{2(x - 6)} = \frac{x^2 - 36}{2(x - 6)}
\]

5. **Factor the numerator:**

The numerator \(x^2 - 36\) is a difference of squares. It factors as:
\[
x^2 - 36 = (x - 6)(x + 6)
\]

6. **Combine and simplify:**

Substitute back, we get:
\[
\frac{(x - 6)(x + 6)}{2(x - 6)}
\]

As long as \(x \neq 6\), we can cancel out \(x - 6\):
\[
\frac{x + 6}{2}
\]

Therefore, the simplified answer is:
\[
\frac{x + 6}{2}
\]

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