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Factoring quadratics with difference of squares
- [Voiceover] We're told that the quadratic expressions m squared minus 4m minus 45, and 6m squared minus 150, share a common binomial factor. What binomial factor do they share? And like always pause the video and see if you can work through this. All right, now let's work through this together and the way I am going to do this is I'm just going to try and factor both of them into the product of binomials and maybe some other things and see if we have any common binomial factors. So first let's focus on m squared minus 4m minus 45. So let me write it over here, m squared minus 4m minus 45. So when you're factoring a quadratic expression like this, where the coefficient on the, in this case, m squared term, on the second degree term is one, we could factor it as being equal to m plus a, times m plus b, where a plus b is going to be equal to this coefficient right over here, and a times b is going to be equal to this coefficient right over here. So let's be clear, so, a, let me see another color, so a plus b needs to be equal to negative 4, a plus b needs to be equal to negative 4, and then a times b needs to be equal to negative 45. A times b is equal to negative 45. Now I like to focus on the a times b and think about, well, what could a and b be to get to negative 45? Well if I'm taking the product of two things and if the product is negative that means that they are going to have different signs and if when we add them we get a negative number that means that the negative one has a larger magnitude. So let's think about this a little bit. So a times b is equal to negative 45. So this could be, let's try some values out. So, 1 and 45, those are too far apart. Let's see. 3 and 15, those still seem pretty far apart. Let's see, it looks like 5 and 9 seem interesting. So if we say, if we say 5 times, if we were to say, 5 times negative 9, that indeed is equal to negative 45, and 5 plus negative 9 is indeed equal to negative 4. So a could be equal to 5 and b could be equal to negative 9. And so if we were to factor this, this is going to be m plus 5, times m, I could say m plus negative 9, but I'll just write m minus 9. So just like that I've been able to factor this first quadratic expression right over there as a product of two binomials. So now let's try to factor the other quadratic expression. Let's try to factor 6m squared minus 150. And let's see, the first thing I might want to do is, both 6m squared and 150, they're both divisible by 6. So let me write it this way, I could write it as, 6m squared minus 6 times, let's see, 6 goes into 150, 25 times. So all I did is I rewrote this and really I just wrote 150 as 6 times 25. And now you can clearly see that we can factor out a 6. You can view this as undistributing the 6. So this is the same thing as 6 times m squared minus 25, which we recognize this is a difference of squares. So it's all going to be 6 times, m plus 5, times m minus 5. And so we've factored this out as a product of binomials and a constant factor here, 6, and so, what is their shared, common or what is their common binomial factor that they share? Well you see when we factored it out, they both have an m plus 5. So m plus 5 is the binomial factor that they share.