Subtract the rational expressions x^2/2x−12−18/x−6. Simplify the answer.
Responses
x+6/2
the fraction with numerator x plus 6 and denominator 2 -this is the correct answer.
x−6/2
the fraction with numerator x minus 6 and denominator 2 -this is the incorrect answer.
x^2−18/x−6
the fraction with numerator x squared minus 18 and denominator x minus 6 -this is the incorrect answer.
x^2−36/2(x−6) -this is the incorrect answer.
explain how you would get that as your correct answer x+6/2.
To subtract the rational expressions x^2/2x−12 and 18/x−6, we need to find a common denominator. The common denominator is (x-6)(2x-12).
Rewriting the expressions with the common denominator, we have:
x^2/2x−12 = x^2(x-6)/(2x-12)(x-6)
18/x−6 = 18(2x-12)/(2x-12)(x-6)
Now, we can subtract the two expressions:
x^2(x-6)/(2x-12)(x-6) - 18(2x-12)/(2x-12)(x-6) = (x^2(x-6) - 18(2x-12))/(2x-12)(x-6)
Expanding the numerator, we have:
(x^3 - 6x^2 - 36x + 216 - 36x + 216)/(2x-12)(x-6)
(x^3 - 6x^2 - 72x + 432)/(2x-12)(x-6)
Simplifying the numerator, we get:
(x^3 - 6x^2 - 72x + 432)/(2x-12)(x-6) = ((x+6)(x^2 - 12))/(2x-12)(x-6)
Now, we can simplify the expression by canceling out the common factors:
((x+6)(x^2 - 12))/(2x-12)(x-6) = (x+6)/(2(x-6))
Therefore, the simplified answer is (x+6)/2.