Simplifying (x² - 6x + 4) / ((x² - 4)(x - 2))

[danni7070]

How do I simplify this?

$$\frac{1}{x-2} - \frac{1}{x^2-4} = \frac{(x^2-4)-(x-2)}{(x^2-4)(x-2)}$$

 

[Dick]

Combine the 4 and 2 in the numerator so you have a single quadratic. Then see if it factors so you can see of if there are any common factors you can cancel.

 

[danni7070]

$$\frac{x^2-x-2}{(x^2-4)(x-2)}$$

I know I should be able to see this instantly but I'm not! What comes next?

 

[Dick]

You don't have to see it instantly. Can you factor the numerator?

 

[danni7070]

Like this?

$$\frac{x^2-x-2}{(x-2)(x+2)(x-2)}$$

 

[danni7070]

$$\frac{x^2-x-2}{(x-2)^2(x+2)} = \frac{x+1}{(x-2)(x+2)}$$

The final step is the answer but I don't know how to get there. What are the rules here?

 

[salman213]

Like this?

$$\frac{x^2-x-2}{(x-2)(x+2)(x-2)}$$

after this stage

factor the numerator

x2 - x - 2 = (x-2)(x+1)

right?

then u can cancel t he (x-2) in the numerator and denominator and ur left with that answer that ur looking for

 

[danni7070]

I'm having one of those dumb days right know. Jesus!

Thank you Dick and salman213.

 

[HallsofIvy]

Like this?

$$\frac{x^2-x-2}{(x-2)(x+2)(x-2)}$$

They said factor the numerator! (Although factoring the denominator doesn't hurt.)