How do you solve #x^2/(x^2-4) = x/(x+2)-2/(2-x)#?

Question

Savannah Anderson · Accepted Answer

There is no solution.

First rearrange the equation so that a LCM of the denominator can be obtained. #x^2/(x^2-4)=x/(x+2)-2/(2-x)# #x^2/(x^2-4)=x/(x+2)-2/(-1*(x-2))# #x^2/(x^2-4)=x/(x+2)+2/(x-2)# Now, multiply both sides by #(x-2)(x+2)#, since this is the LCM. Doing this gets: #x^2=x(x-2)+2(x+2)# #x^2=x^2-2x+2x+4# #0=4# which is not possible. There is no solution.

Kennedy Adams · Answer

undefined To solve the equation x^2/(x^2-4) = x/(x+2) - 2/(2-x), we can follow these steps:

1. Start by finding a common denominator for the fractions on the right side of the equation. The common denominator is (x+2)(2-x).

2. Rewrite the fractions on the right side with the common denominator:
   x/(x+2) = x(2-x)/[(x+2)(2-x)]
   -2/(2-x) = -2(x+2)/[(x+2)(2-x)]

3. Simplify the fractions on the right side:
   x/(x+2) = (2x-x^2)/[(x+2)(2-x)]
   -2/(2-x) = -2(x+2)/[(x+2)(2-x)]

4. Combine the fractions on the right side:
   (2x-x^2)/[(x+2)(2-x)] - 2(x+2)/[(x+2)(2-x)]

5. Now, we can eliminate the denominators by multiplying both sides of the equation by [(x+2)(2-x)]:
   x^2 - 2x - x^2 - 4x - 4 = 2x - x^2 - 4x - 8

6. Simplify the equation:
   -6x - 4 = -2x - 8

7. Combine like terms:
   -6x + 2x = -8 + 4
   -4x = -4

8. Divide both sides by -4:
   x = 1

Therefore, the solution to the equation x^2/(x^2-4) = x/(x+2) - 2/(2-x) is x = 1.