Solving Rational Equations?

Question

How do you do this problem, I know you have to find the common denominator but I'm not sure how to.

x/x-2 + x/x^2-4 = x+3/x+2

Answer 1

The common denominator is (x + 2)(x - 2), since x^2 - 4 is the difference of two squares which is factored as (x + 2)(x -2).

You have to multiply the first term by (x + 2) and multiply the third term by (x - 2). The middle term already has the common denominator.

The equation looks like this:

[x(x+2) + x] / [(x + 2)(x - 2)] = [(x + 3)(x - 2)]/[(x+2)(x - 2)]

Simplify:

(x^2 + 2x + x)/ [(x + 2)(x -2)] = [(x+3)(x -2)]/[(x+2)(x - 2)]

You can factor the left side and subtract the right side from the left:
[x(x + 3)] - [x +3)(x - 2)]/[(x + 2)(x - 2)] = 0

(x+3)[x - (x - 2)]/[(x+2)(x - 2)] = 0

Simplify, and multiply both sides by (x-2)(x + 2)
[x+3](-2) = 0

-2x - 6 = 0

-2x = 6

x = -3

Answer 2

there's a short way and a long way. we're going to do it the short way.
look at the 3 denominators:
x-2, x^2-4, and x+2.
lets take the first one and the third one and multiply them out:
(x-2)(x+2) = x^2+2x-2x-4 = x^2-4
so we see that the middle denominator is the multiplication of the other 2.

as you know, you can multiply any term by 1 and not change it's value. we're going to use this to our advantage.

above, we multiplied the first denominator (x-2) by (x+2). now, as said, we can't just multiply the first term by 1/x+2 because that would change the value - but we CAN multiply by (x+2)/(x+2) because that just equals 1.

so we'll multiply the first term by (x+2)/(x+2) and get x(x+2)/(x^2-4).

we'll multiply the second term by 1/1 because it already has the right denominator.

and the third term by (x-2)/(x/2) and get (x+3)(x-2)/(x^2-4).

so, after multiplying it all out, we get:
x^2 + 2x + x = x^2 + 3x -2x - 6 (we can ignore the denominators now because they are all the same!).
combine terms
x^2 + 3x = x^2 + x - 6
2x = -6
x=-3

Answer 3

okay, first you want to fact (x^2 - 4) to (x +2) and (x-2).

So now you've got

x/x-2 + x/(x+2)(x-2) = x + 3/x +2

Now you have a common denominator (x^2-4) and you multiply everything else in the equation by whatever is needed to give that denominator

x/x-2 X (x+2/x+2) = (x^2 + 2x)/x^2 -4
The second you leave alone, as it already has the common denominator

x X (x^2-4/x^2-4) = x^3 - 4x/x^2 -4

3/x + 2 X (x-2/x-2) = 3x -6/(x^2 -4)

So your whole equation looks like this:

[(x^2 + 2x)/x^2 -4] + [x/x^2-4] = [ x^3 - 4x/x^2 -4] + [3x -6/(x^2 -4)]

Now, combine your equations

(x^2 + 3x)/(x^2 - 4) = (x^3 - x -6)/(x^2 -4)

And solve from there.