Simplify (2x)/(x^2-6x+9)-1/(x+1)-8/(x^2-2x-3)

Problem

(2*x)/(x2−6*x+9)−1/(x+1)−8/(x2−2*x−3)

Solution

1. Factor the denominators of each fraction to identify the least common denominator.

x2−6*x+9=(x−3)2

x2−2*x−3=(x−3)*(x+1)

2. Rewrite the expression using the factored forms.

(2*x)/((x−3)2)−1/(x+1)−8/((x−3)*(x+1))

3. Identify the least common denominator (LCD), which is (x−3)2*(x+1)

4. Adjust each fraction to have the LCD by multiplying the numerator and denominator by the missing factors.

(2*x*(x+1))/((x−3)2*(x+1))−((x−3)2)/((x−3)2*(x+1))−(8*(x−3))/((x−3)2*(x+1))

5. Expand the numerators.

2*x*(x+1)=2*x2+2*x

(x−3)2=x2−6*x+9

8*(x−3)=8*x−24

6. Combine the numerators over the common denominator, being careful to distribute the negative signs.

((2*x2+2*x)−(x2−6*x+9)−(8*x−24))/((x−3)2*(x+1))

7. Simplify the numerator by combining like terms.

2*x2−x2+2*x+6*x−8*x−9+24

x2+15

Final Answer

(2*x)/(x2−6*x+9)−1/(x+1)−8/(x2−2*x−3)=(x2+15)/((x−3)2*(x+1))

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