Simplify x/(x^2+x-2)-2/(x^2-5x+4)

Problem

x/(x^2+x−2)−2/(x^2−5*x+4)

Solution

1. Factor the denominators of both fractions to find the least common denominator.

x^2+x−2=(x+2)*(x−1)

x^2−5*x+4=(x−4)*(x−1)

2. Rewrite the expression using the factored denominators.

x/((x+2)*(x−1))−2/((x−4)*(x−1))

3. Identify the least common denominator (LCD), which is (x+2)*(x−1)*(x−4)

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

(x*(x−4))/((x+2)*(x−1)*(x−4))−(2*(x+2))/((x+2)*(x−1)*(x−4))

5. Combine the numerators over the common denominator.

(x*(x−4)−2*(x+2))/((x+2)*(x−1)*(x−4))

6. Expand the terms in the numerator.

(x^2−4*x−2*x−4)/((x+2)*(x−1)*(x−4))

7. Simplify the numerator by combining like terms.

(x^2−6*x−4)/((x+2)*(x−1)*(x−4))

Final Answer

x/(x^2+x−2)−2/(x^2−5*x+4)=(x^2−6*x−4)/((x+2)*(x−1)*(x−4))

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