To subtract the rational expressions \(\frac{x^2}{2x - 12} - \frac{18}{x - 6}\), we first need to simplify and find a common denominator.
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Factor the denominators:
The first denominator can be factored as: \[ 2x - 12 = 2(x - 6) \] So the expression becomes: \[ \frac{x^2}{2(x - 6)} - \frac{18}{x - 6} \]
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Find a common denominator:
The common denominator will be \(2(x - 6)\).
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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: \[ \frac{18}{x - 6} = \frac{18 \cdot 2}{(x - 6) \cdot 2} = \frac{36}{2(x - 6)} \]
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Now, subtract the fractions:
Now we can subtract: \[ \frac{x^2}{2(x - 6)} - \frac{36}{2(x - 6)} = \frac{x^2 - 36}{2(x - 6)} \]
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Factor the numerator:
The numerator \(x^2 - 36\) is a difference of squares. It factors as: \[ x^2 - 36 = (x - 6)(x + 6) \]
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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: \[ \text{Start Fraction x plus 6 over 2 End Fraction} \]