 There's a couple of special problems that show up when we look at factoring quadratics So for example, let's try to factor 8x squared minus 16x minus 64 So if we try to use the quadratic formula on the equation 8x squared minus 16x minus 64 equals zero We start out by writing something like this, which gives us unpleasantly large numbers Now if we wanted to we could grind our way through this and it would be good for building character tenacity Perseverance, but it would also be a lot of extra effort So a useful idea to keep in mind is to try the easy things first And in this case one of the easy things is to recognize that every term has a factor of 8 and so we can remove this common factor and And now we can try to factor x squared minus 2x minus 8 instead of our original polynomial So we'll solve x squared minus 2x minus 8 equals zero using the quadratic formula And we'll get our solutions 4 or negative 2 Which gives us the factors x minus 4 x plus 2 Or how about something like this 4x squared minus 7x plus 12 So we'll solve 4x squared minus 7x plus 12 equals zero using the quadratic formula So we'll substitute our coefficients and do a little bit of arithmetic and Before we go on it's useful to keep in mind two important ideas First computers can't think they have to follow steps until the end no matter how far away that end is And the only reason that it works for computers is they can follow steps very quickly very accurately on the other hand humans can and should think and what that means is you can stop when you see the solution in This case while the quadratic formula requires us to subtract We can look at this 49 minus 192 and immediately recognize that that's going to be a negative number and our Solution will not be a real number, so we don't actually have to do the subtraction We can stop here and say that we have no real solutions and that's important because we want to find factors that are rational and a non real factor isn't rational and so we might say that since 4x squared minus 7x plus 12 equals zero has no Real solutions our polynomial cannot be factored We can run into a similar problem with x squared minus 3x minus 12 so solving x squared minus 3x minus 12 equals zero using the quadratic formula and we get down to this point and Once again the important idea here is that computers can't think people should and The thing to notice here 37 is not a perfect square, so whatever the solutions are they are not going to be rational So the corresponding factors won't be rational either and so again We can conclude since our equation has no rational solutions our polynomial has no rational factors