Fractions involving Quadratic Terms: Difference of Squares

Fractions involving Quadratic Terms Difference of Squares
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You can see that we’ve got a quadratic on both numerator and denominator. So let’s go factorize those individually. I’m going to start with the numerator use your usual cross method x and x. I know that’s going to be negative and negative because that’s positive and that’s negative, so negative negative is what I need.

So I’m gonna use negative 3 and the other one will probably be negative 4. So I’ll start with this. x times negative 3 is negative 3x, so the other one is negative 4, x times negative 4 is negative 4x, and you can see that negative 3x minus 4x is negative 7x which is what we want, so we’ve got the right one.

Guys, you should be able to tell me how we’re going to factorize the denominator. You can see that 16 is 4 squared, so it’s going to be difference of two squares. So the top will be x minus 3x minus 4, the denominator as I said 16 is 4 squared. So difference of two squares. How do we factorize that? x plus 4 x minus 4. Every single one of you should be able to do this. So now we can start seeing if we can cancel anything out and you can see that x minus 4 is common, so this is the favorite part. So cross them out. We’ve just got x minus 3 over x plus 4 left. Okay? So it’s pretty much the same thing but you just need to do a little bit more factorizations of quadratics. Let’s go for it, again, now x and x for the sorry the numerator. Now it’s negative, so

I’m going to use positive and negative. I’ll use a positive 7 first of all. So that’s going to be 7x and then the other one will be 13 because 7 times 13 makes 91, so you can use a bit of your calculator if you like the track. Now that’s going to be negative because I use positive there and cross negative 13. And you can see that 7x minus 13x is negative 6x which is that. So we’ve got the right form. So that’s the numerator, um okay.

Now the denominator I’ll put that in in a second on the numerator. But what I did for the denominator is see how this it’s negative x squared plus 49. I just swap them around because I always want my x square to come first. And as I said I don’t want to have a negative in front of my x squared, so I factorize by just negative or negative 1 basically. And then that will be positive x squared and that will change to a negative 49. And the reason why I did that is because I can see that it’s going to become an x squared minus 49 which is 7 squared, isn’t it? 49 is 7 squared and I just replaced x plus 7 and x minus 13 for the numerator because I’ve just done the quadratic.

Okay now let’s factorize the denominator guys all of you should do this. You can see that x minus 7 x plus 7. Difference of two squares. And you can see that I can cross these out. Cancel them, so I just have that left. Now I don’t really want to leave. That looks pretty ugly. So I just kind of expanded it out. Negative x plus seven. If it’s just a negative, you might as well just expand it. Just so it looks a little bit more neat.


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