Question

Subtract the rational expressions. x2/2x-12 - 18/x-6

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:
\[
2x - 12 = 2(x - 6)
\]
So we rewrite the expression:
\[
\frac{x^2}{2(x - 6)} - \frac{18}{x - 6}
\]

2. **Find a common denominator:**

The common denominator between \(2(x - 6)\) and \(x - 6\) is \(2(x - 6)\).

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

The first term is already using the common denominator:
\[
\frac{x^2}{2(x - 6)}
\]

The second term needs to be adjusted by multiplying its 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 two fractions:**
\[
\frac{x^2}{2(x - 6)} - \frac{36}{2(x - 6)} = \frac{x^2 - 36}{2(x - 6)}
\]

5. **Factor the numerator:**

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

So we rewrite the expression:
\[
\frac{(x - 6)(x + 6)}{2(x - 6)}
\]

6. **Cancel the common terms:**

Since \(x - 6\) is present in both the numerator and denominator (as long as \(x \neq 6\)), we can cancel it:
\[
\frac{x + 6}{2} \quad (x \neq 6)
\]

Thus, the final result is:
\[
\frac{x + 6}{2} \quad (x \neq 6)
\]