how to solve x/x^2-4+1/x^2+2x

Question

This is an algebra I addition problem x divided by x squared minus four PLUS 1 divided by x squared plus two x

 

        x                      +                    1

x^2 - 4                                       x^2 + 2x

Answer 1

Begin by factoring the denominators:

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

The common denominator is x(x + 2)(x - 2)

Multiply the first fraction by x/x and the second fraction by (x -2)/(x - 2)

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

Multiply the numerators and the denominators to get

x²/[x(x - 2)(x + 2)] + (x - 2)/[x(x - 2)(x + 2)]

Now that the denominators are the same, we can add the numerators.  We should get

(x² + x - 2)/[x(x - 2)(x + 2)]

After factoring the numerator, we have

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

Since both the numerator and the denominator have an x + 2 term, we can cancel them out to get

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

You can distribute the x in the denominator to get a final answer of

(x - 1)/(x² - 2x)