Adding & subtracting rational expressions: like denominators | High School Math | Khan Academy

So let's add six over two x squared minus seven to negative three x minus eight over two x squared minus seven. And like always, pause the video and try to work it out before I do.

Well, when you look at this, we have these two rational expressions and we have the same denominator, 2x squared minus 7. So you could say we have 6 over 2x squared minus 7, and then we have negative 3x minus 8 over 2x squared minus 7. That is one way to think about it.

So if you have the same denominator, this is going to be equal to our denominator, which is going to be 2x squared minus 7. Then we just add the numerators, so it's going to be 6 plus negative 3x, which is negative 3x minus 8.

If we want to simplify this a little bit, we'd recognize that we could add these two constant terms, the 6 and the negative 8. 6 plus negative 8 is going to be negative 2, so it's going to be negative 2. Adding a negative 3x is the same thing as subtracting 3x, so negative 2 minus 3x.

All of that, in that same blue color, is over 2x squared minus 7. And we're done. We've just added these two rational expressions.

Let's do another example. So here we want to subtract one rational expression from another. See if you can figure that out.

Well, once again, both of these rational expressions have the exact same denominator. The denominator for both of them is 14x squared minus 9. So the denominator of the difference, I guess we can call it that, is going to be 14x squared minus 9.

Did I say 4x squared before? 14x squared minus 9. That's the denominator for both of them, so that's going to be the denominator of our answer right over here.

We can just subtract the numerators. So we're going to have 9x squared plus 3 minus, all of this business, minus negative 3x squared plus 5. We can distribute the negative sign. This is going to be equal to 9x squared plus 3.

If you distribute the negative sign, the negative of negative 3x squared is going to be plus 3x squared, and the negative of positive 5 is going to be negative 5. So we're going to subtract 5 from that, and all of that is going to be over 14x squared minus 9.

In the numerator, we can do some simplification. We have 9x squared plus 3x squared, so that's going to be equal to 12x squared. Then we have 3 plus negative 5 or we could say 3 minus 5, which is going to be negative 2.

All of that is going to be over 14x squared minus 9. And we're all done. We have just subtracted, and we could think about whether there’s any way to simplify this more. Are there any common factors?

But these both could be considered differences of squares, but they're going to be differences of squares of different things, so they're not going to have common factors. So this is about as simple as we can get.