Question

How do you change the fraction `(w-3)/(w+5)` into an equivalent fraction with the denominator `w^2 + w - 20`?

Answer 1

multiply it by `{w -4}/{w -4}` to get `{(w-3)(w-4)}/{w^2 + w - 20}`

Explanation:

multiply it by `{w -4}/{w -4}`

Basically you want to solve the equation `(w+5)x = w^2 + w - 20`

As you can clearly see, you will be left with a quadratic expression as the denominator. Therefore `x` should be in the form `(w-a)` where a is some integer. So that leaves us with the equation

`(w+5)(w-a) = w^2 + w - 20`

We can factorize the right hand side to `(w+5)(w-4)` and that makes it pretty clear that a is 4 as `(w+5)(w-a)=(w+5)(w-4)`.
Hence, we need to multiply the fraction by `{w -4}/{w -4}`. Of course, since `{w -4}/{w -4}=1` the value of the expression will not change.