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

1 answer

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: \[ \text{Start Fraction x plus 6 over 2 End Fraction} \]