Question

How do you subtract `\frac{2x}{x^2+10x+25}-\frac{3x}{2x^2+7x-15}`?

Answer 1

First, we're going to simplify the expressions a bit. We can do this by factoring both the denominators:
`x^2+10x+25 = (x+5)^2`

`2x^2+7x-15=(x+5)(2x-3)`

To substract two fractions with a different denominator, we will always have to find the LCM of the denominators. In this case it is `(x+5)^2(2x-3)` since this can evenly be divided by both the denominators.

Let's combine what we've got already:
`(2x)/(x+5)^2-(3x)/((x+5)(2x-3))`

Now for changing to the LCM:
`(2x-3)/(2x-3)*(2x)/(x+5)^2 - (x+5)/(x+5)(3x)/((x+5)(2x-3))`

`= (2x*(2x-3)-3x*(x+5))/((x+5)^2(2x-3))`

Distribute the terms:

`(4x^2-6x-3x^2-15x)/((x+5)^2(2x-3))`

`= (x^2-21x)/((x+5)^2(2x-3))`

You could've also done this without factoring, but it would've been way harder. Factoring always makes your life easier!

I really hope this helped.