Question

How do you simplify `(2t)/(s^2 - st) + (s+t)/(t^2 - s^2)`?

Answer 1

`(s - 2t)/(s(t-s))`

Explanation:

Your starting expression looks like this

`(2t)/(s^2 - st) + (s + t)/(t^2 - s^2)`

Notice that you can write

`s^2 - st = - (st - s^2)`

The expression becomes

`-(2t)/(st - s^2) + (s + t)/(t^2 - s^2)`

Now focus on factoring the two denominators. For `st - s^2`, you can use `s` as a common factor to write

`st - s^2 = s * (t - s)`

Notice that the second denominator is a difference of squares, which means that you can factor it using

`color(blue)(a^2 - b^2 = (a-b)(a+b))`

In your case, you would have

`t^2 - s^2 = (t-s)(t+s)`

The expression can thus be written as

`-(2t)/(s(t-s)) + color(red)(cancel(color(black)((t+s))))/((t-s) color(red)(cancel(color(black)((t+s)))))`

`-(2t)/(s(t-s)) + 1/(t-s)`

Multiply the second fraction by `1 = s/s` to get

`-(2t)/(s(t-s)) + s/(s(t-s)) = color(green)((s - 2t)/(s(t-s))`