Solve for x x/(x-4)-1/(x+3)=28/(x^2-x-12)

Problem

x/(x−4)−1/(x+3)=28/(x2−x−12)

Solution

1. Factor the denominator on the right side of the equation to find the least common denominator.

x2−x−12=(x−4)*(x+3)

2. Identify the restricted values for x by setting the denominators to zero.

x≠4,x≠−3

3. Multiply every term in the equation by the least common denominator (x−4)*(x+3) to clear the fractions.

x*(x+3)−1*(x−4)=28

4. Distribute the terms and simplify the left side of the equation.

x2+3*x−x+4=28

x2+2*x+4=28

5. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 28 from both sides.

x2+2*x−24=0

6. Factor the quadratic equation to solve for x

(x+6)*(x−4)=0

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

x+6=0⇒x=−6

x−4=0⇒x=4

8. Check the solutions against the restricted values identified in step 2. Since x=4 is a restricted value (it makes the original denominators zero), it is an extraneous solution.

x=−6

Final Answer

x/(x−4)−1/(x+3)=28/(x2−x−12)⇒x=−6

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