Adding & subtracting rational expressions: like denominators

- [Voiceover] So let's add six over two X squared minus seven to negative 3 X minus eight over two X squared minus seven. And like always, pause the video and try to work it out before I do. When you look at this, we have these two rational expressions and we have the same denominator, two X squared minus seven. So you could say, we have six two X squared minus sevenths and then we have negative three X minus eight two X squared minus sevenths is one way to think about it. So if you have the same denominator, this is going to be equal to, this is going to be equal to... our denominator is going to be two X squared minus seven, two X squared minus seven, and then we just add the numerators. So it's going to be six plus negative three X, negative three X minus eight. So if we want to simplify this a little bit, we'd recognize that we can add these two constant terms, the six and the negative eight. Six plus negative eight is going to be negative two, so it's going to be negative two and then adding a negative three X, that's the same thing as subtracting three X, so negative two minus three X, all of that over, all of that with that same blue color, all of that over two X squared minus seven. 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. So 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 14 X squared minus nine, 14 X squared minus nine. So the denominator of the difference, I guess we can call it that, is going to be 14 X squared minus nine. So 14 X squared minus nine. Did I say four X squared before? 14 X squared minus nine, that's the denominator of both of them, so that's going to be the denominator of our answer right over here. And so, we can just subtract the numerators. So we're gonna have nine X squared plus three minus all of this business, minus negative three X squared plus five. And so we can distribute the negative sign. This is going to be equal to nine X squared plus three, and then, if you distribute the negative sign, the negative of negative three X squared is going to be plus three X squared and then the negative of positive five is going to be negative five, so we're gonna subtract five from that, and all of that is going to be over 14 X squared minus nine. 14 X squared minus nine. And so in the numerator we can do some simplification. We have nine X squared plus three X squared, so that's going to be equal to 12 X squared. And then, we have... we have three plus negative five, or we can say three minus five, so that's going to be negative two, and all of that is going to be over 14 X squared minus nine. 14 X squared minus nine. And we're all done. We have just subtracted. And we can think about it, is there any way we can 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.