Question

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

x+62
Start Fraction x plus 6 over 2 End Fraction

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

x−62
Start Fraction x minus 6 over 2 End Fraction

x2−362(x−6)

Answer 1

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**.