Question

How do simplify `\frac { a - 4} { a ^ { 2} - 2a - 8} \div \frac { 4a - 20} { a - 5}`?

Answer 1

`1/(4a+8)` where `a!=4, -2, or 5`

Explanation:

`(a-4)/(color(red)(a^2-2a-8))-:(color(blue)(4a-20))/(a-5)`

Start by factoring all factorable expressions

`color(red)(a^2-2a-8=(a-4)(a+2)` using reverse FOIL, diamond method, etc.

`color(blue)(4a-20=4(a-5))` using reverse distributive property

So we now have:

`(a-4)/(color(red)((a-4)(a+2)))-:(color(blue)(4(a-5)))/(a-5)`

Now would be a good time to point out that `a!=4, -2, or 5` because that would mean we would have to divide by 0 which is not allowed.

Then we divide away expressions that are the same on the top and bottom:

`cancel(a-4)/(color(red)((cancel(a-4))(a+2)))-:(color(blue)(4cancel((a-5))))/cancel((a-5))`

Resulting in:

`1/(a+2)-:4/1=1/(a+2)xx1/4=1/(4(a+2))=1/(4a+8)`