Question

subtract the rational expressions \\(\frac{x^2}{2x - 12} - \frac{18}{x - 6}\\). simplify the answer (1 point) \\(\frac{x^2 - 36}{2(x - 6)}\\) \\(\frac{x^2 - 18}{x - 6}\\) \\(\frac{x - 6}{2}\\) \\(\frac{x + 6}{2}\\)

Explanation:

Step1: Factor the denominator

First, factor the denominator of the first fraction. We have \(2x - 12 = 2(x - 6)\). So the expression becomes \(\frac{x^2}{2(x - 6)}-\frac{18}{x - 6}\).

Step2: Find a common denominator

The common denominator for the two fractions is \(2(x - 6)\). To get the second fraction to have this common denominator, we multiply the numerator and denominator of \(\frac{18}{x - 6}\) by 2. This gives us \(\frac{18\times2}{2(x - 6)}=\frac{36}{2(x - 6)}\). Now the expression is \(\frac{x^2}{2(x - 6)}-\frac{36}{2(x - 6)}\).

Step3: Subtract the numerators

Since the denominators are now the same, we can subtract the numerators: \(\frac{x^2 - 36}{2(x - 6)}\). Notice that \(x^2 - 36\) is a difference of squares, which factors as \((x - 6)(x + 6)\). So we have \(\frac{(x - 6)(x + 6)}{2(x - 6)}\).

Step4: Cancel common factors

We can cancel out the common factor of \((x - 6)\) from the numerator and the denominator (assuming \(x
eq6\) to avoid division by zero). This leaves us with \(\frac{x + 6}{2}\). Wait, but let's check our factoring again. Wait, \(x^2-36=(x - 6)(x + 6)\)? Wait, no, \(x^2 - 36=(x - 6)(x + 6)\) is correct, but wait, when we subtract \(\frac{x^2}{2(x - 6)}-\frac{36}{2(x - 6)}=\frac{x^2 - 36}{2(x - 6)}\). But \(x^2-36=(x - 6)(x + 6)\)? Wait, no, \(x^2 - 36=(x - 6)(x + 6)\) is correct, but if we have \(x^2-36=(x - 6)(x + 6)\), then \(\frac{(x - 6)(x + 6)}{2(x - 6)}=\frac{x + 6}{2}\) when \(x
eq6\). Wait, but let's check the original problem again. Wait, maybe I made a mistake in the subtraction. Wait, the first term is \(\frac{x^2}{2(x - 6)}\) and the second term is \(\frac{18}{x - 6}\), so when we get a common denominator, it's \(\frac{x^2}{2(x - 6)}-\frac{36}{2(x - 6)}=\frac{x^2 - 36}{2(x - 6)}\). But \(x^2-36=(x - 6)(x + 6)\)? Wait, no, \(x^2 - 36=(x - 6)(x + 6)\) is correct, but if we have \(x^2-36=(x - 6)(x + 6)\), then canceling \((x - 6)\) gives \(\frac{x + 6}{2}\). Wait, but let's check with another approach. Let's plug in a value for \(x\), say \(x = 0\). Original expression: \(\frac{0^2}{2(0 - 12)}-\frac{18}{0 - 6}=\frac{0}{-24}-\frac{18}{-6}=0 + 3 = 3\). Now check the options:

• Option 1: \(\frac{0^2 - 36}{2(0 - 6)}=\frac{-36}{-12}=3\). Wait, no, option 1 is \(\frac{x^2 - 36}{2(x - 6)}\), but when \(x = 0\), that's 3. But our simplified form was \(\frac{x + 6}{2}\), when \(x = 0\), that's \(\frac{6}{2}=3\), which matches. Wait, but let's check option 4: \(\frac{x + 6}{2}\), when \(x = 0\), it's 3, which matches. Wait, but let's check \(x = 7\). Original expression: \(\frac{49}{2(1)}-\frac{18}{1}=\frac{49}{2}-18=\frac{49 - 36}{2}=\frac{13}{2}\). Option 4: \(\frac{7 + 6}{2}=\frac{13}{2}\), which matches. Option 3: \(\frac{7 - 6}{2}=\frac{1}{2}\), which doesn't match. So the correct answer is \(\frac{x + 6}{2}\), which is option 4. Wait, but let's go back to the algebra. Wait, maybe I made a mistake in the factoring. Wait, \(x^2 - 36=(x - 6)(x + 6)\), so \(\frac{(x - 6)(x + 6)}{2(x - 6)}=\frac{x + 6}{2}\) (for \(x

eq6\)). So the correct answer is \(\frac{x + 6}{2}\), which is the fourth option.

Answer:

\(\frac{x + 6}{2}\) (corresponding to the option with \(\frac{x + 6}{2}\))