To subtract the rational expressions \( \frac{x^2}{2x - 12} - \frac{18}{x - 6} \), we first need to simplify the denominators.
Notice that \( 2x - 12 \) can be factored as:
\[ 2x - 12 = 2(x - 6) \]
Thus, we rewrite the first expression:
\[ \frac{x^2}{2(x - 6)} - \frac{18}{x - 6} \]
Now, our expression is:
\[ \frac{x^2}{2(x - 6)} - \frac{18}{x - 6} \]
Next, to combine these fractions, we need a common denominator. The least common denominator (LCD) between \( 2(x - 6) \) and \( (x - 6) \) is \( 2(x - 6) \).
We rewrite the second fraction with the common denominator:
\[ \frac{18}{x - 6} = \frac{18 \cdot 2}{2(x - 6)} = \frac{36}{2(x - 6)} \]
Now, our expression becomes:
\[ \frac{x^2}{2(x - 6)} - \frac{36}{2(x - 6)} \]
Combine the fractions:
\[ \frac{x^2 - 36}{2(x - 6)} \]
Next, notice that \( x^2 - 36 \) is a difference of squares, which can be factored:
\[ x^2 - 36 = (x - 6)(x + 6) \]
Substituting this back into our expression gives:
\[ \frac{(x - 6)(x + 6)}{2(x - 6)} \]
Now we can simplify this expression by canceling the \( (x - 6) \) term:
\[ \frac{x + 6}{2} \]
Thus, the final simplified answer is:
\[ \frac{x + 6}{2} \]
From your provided responses, this corresponds to:
Start Fraction x plus 6 over 2 End Fraction.