Solve for x (x+5)/(x-2)=5/(x+2)+28/(x^2-4)

Problem

(x+5)/(x−2)=5/(x+2)+28/(x2−4)

Solution

1. Factor the denominators to find a common denominator.

x2−4=(x−2)*(x+2)

2. Identify the least common denominator (LCD) and the restricted values for x

LCD=(x−2)*(x+2)

x≠2,x≠−2

3. Multiply every term in the equation by the LCD to clear the fractions.

(x−2)*(x+2)⋅(x+5)/(x−2)=(x−2)*(x+2)⋅5/(x+2)+(x−2)*(x+2)⋅28/((x−2)*(x+2))

4. Simplify the resulting equation by canceling common factors.

(x+2)*(x+5)=5*(x−2)+28

5. Expand both sides of the equation.

x2+7*x+10=5*x−10+28

6. Combine like terms on the right side.

x2+7*x+10=5*x+18

7. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 5*x and 18 from both sides.

x2+2*x−8=0

8. Factor the quadratic equation.

(x+4)*(x−2)=0

9. Solve for x by setting each factor to zero.

x+4=0⇒x=−4

x−2=0⇒x=2

10. Check the solutions against the restricted values identified in step 2.

x=2* is restricted because it makes the denominator zero.

x=−4* is a valid solution.

Final Answer

(x+5)/(x−2)=5/(x+2)+28/(x2−4)⇒x=−4

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