Expert: Steve Holleran - 8/13/2008

Question
Can you help me with the folloqing?

1      +   1           1    -     2x
___        ___     =   ___       _______  

x + 2      2           x-2        x^2 - 4

Answer
Hi Nicole,

Okay, what you want to do here is to factor the x^2-4 denominator on the right so that your equation looks like:

   1 / (x+2) + 1/2 = 1 / (x-2) - 2x /[(x+2)(x-2)]

Now, we need to find the LCD of all the fractions and then multiply each fraction in the equation by that LCD.

If you look at all the denominators, you'll see that the LCD is

                             2(x+2)(x-2)

So, let's multiply each fraction by this:

Left side:

 1/(x+2) * 2(x+2)(x-2)  + 1/2 * 2(x+2)(x-2)

after canceling, you should have

 2(x-2) + (x+2)(x-2) = 2x - 4 + x^2 - 4


Right side:

 1/(x-2) * 2(x+2)(x-2) - 2x/[(x+2)(x-2)] * 2(x+2)(x-2)

= 2x + 4 - 4x  = -2x + 4


Now set left side = right side:

           2x - 4 + x^2 - 4  =  -2x + 4

           x^2 + 2x - 8      =  -2x + 4

           x^2 + 4x - 12     = 0

           (x + 6)(x - 2)    = 0     so x = -6 or x = 2.

But x = 2 has to be thrown out (extraneous) because it make the x-2 denominator in the original problem = 0,  and the only correct answer is x = -6.

Hope this helps
Steve