 One of the reasons that completing the square is extremely useful is that it applies in a number of different cases and One important application is that we can use it to factor anything that is actually factorable Provided we know some things and in particular the great secret of factoring is there really isn't any There's only two methods of factoring that are important The first is removing a common factor from a group of terms So if I have 15x squared minus 3x, I might make the observation Ah ha both of these have factors of 3 and X so I could remove a 3x from both terms and factor that way The other factoring that's really helpful is to recognize when we have the difference of squares I have 4x squared minus 9. I take a look this 4x squared is a perfect square It's the square of 2x 9 is a perfect square It's the square of 3 that subtraction says I'm dealing with a difference of squares And that says I can factor it immediately as the sum and difference of the corresponding square roots Provided I have these two things on hand Anything I can factor can be factored using these two methods as long as we can also complete the square So for example, let's consider this very simple factorization problem x squared minus 2x minus 24 And this assumes that we've made at some point the transition to the algebraic form of completing the square We don't absolutely need it But it will make our lives easier if we don't have to stop in the middle to redraw all of our pictures So we'll split off our constant term and then we'll complete the square on the remaining terms So what does that mean? We'll take my expression x squared minus 24x And I'm going to keep the square and the linear term together and I'm going to split off that minus 24 What I want to do is I want to complete the square here So let's see I can complete the square by adding 1 And if you put something in you have to take it out I can't maintain the equality unless whatever I do I undo Very important thing to keep in mind here, this is not an equation We are not trying to solve an equation here So you cannot think about this as doing the same thing to both sides because it won't work You don't have two sides of this expression You have the one thing and it is identical to this thing here This is not an equation, this is an identity, and I'm only allowed to change one side of the identity So here, x squared minus 2x added 1, but take it out If you put it in, take it out, basic kindergarten rule Alright, so let's take a look at that This set of terms here, x squared minus 2x plus 1, that is actually a perfect square I can do some arithmetic here, 24 minus 1 is 25 And let's see, think, think, think, think, oh great, that's a perfect square So now what I have is I have a perfect square minus another perfect square And that is a difference of squares, that means I can factor it immediately As the difference of the two terms, x minus 1 minus 5, and the sum of the two terms, x minus 1 plus 5 And at this point I have some arithmetic to do Possibly the most complicated part, or very likely the part that we will most often get wrong in these questions This is x minus 1 minus 5, that's x minus 6 x minus 1 plus 5, that's x plus 4 And there's a factorization of this expression Well, let's take a look at something a little bit more complicated, x squared minus 9x plus 18 So again, we'll split off the constant and complete the square on the remaining terms So that plus 18, I'll put that aside for a minute, x squared minus 9x And then something else goes in here to make it a perfect square And if you remember how that works, we take half of our constant square it I'm going to add 9 half squared And if I've put it in, I have to take it out So I've added 9 half squared, I have to take out 9 half squared And after all the dust settles, so I convert the fractions, that's minus 9 fourths And that is a perfect square, excellent I now have a difference of perfect squares And I can factor that as 1 minus the other, 1 plus the other And again, there's a little bit of arithmetic we have to do at the end 9 half minus 3 half, that's minus 6 Minus 9 half plus 3 half, that's going to be minus 3 And there's our factorization Alright, well let's take a look at another example So here's something, 6x squared minus x minus 15 Here, our coefficient of x squared is not equal to 1 And that's going to require us to make some changes when we complete the square So let's go ahead and take a look at that And if you go back to our method of solving equations by completing the square It was necessary that we make our first term, our x squared term, into a perfect square And there's two ways we can deal with that One thing we can do is we could factor out a 6 So there's our first important method of factoring, removing a common factor So I'm going to factor out a 6 and I'll have this expression in here And now I can try to complete the square on this thing Now the downside of that is we're going to have to deal with a whole bunch of fractions And fractions are inevitable But let's see if we can put off running into fractions for as long as possible So another possibility, I need to make sure this thing is a perfect square Well the thing that it's missing is another factor of 6 The x squared isn't the problem The thing that's missing is that factor of 6 And again, basic kindergarten rule You can put it in as long as you take it out So the idea is I want to put in a 6 I'll take out a 6 and I can regroup my factors So that's going to give me 6x squared minus 6x minus 90 inside the parentheses And at this point I'll go ahead and follow the same procedure So I'll split off that 90 I have my 6x squared, I have my x term Now for that x term Remember that's going to be the product of 2 times something Which also includes the coefficient of x So that's going to be 2 times 6 times something And remember that coefficient, we needed that coefficient to be 6 So that something has to be 1 half And so I have 6x squared minus 2 times 6 times 1 half times x minus 90 Now again, I'll split off the 90 And I'll try to complete the square on the first two terms So let's see, I want to add the square of this 1 half So I'll add 1 half squared I'll let all the dust settle That's minus 361 over 4 I'll recognize this 361 over 4 is 19 halves And this is 6x minus 1 half squared And again, I have my difference of perfect squares And I can write that as the first one minus the second one First one plus the second one And I have to do a little bit of arithmetic And there's our factorization Now this might not look to be in the form that we're used to seeing Because we have a fraction out here So we can do something about that Again, going back to the first important method of factoring Notice that everything in here has a factor of 2 Everything in here has a factor of 3 And so I can remove those common factors of 2 and 3 And guess what? 2, 3, and the 1, 6 All cancel out, and so I'm left with the factorization 3x minus 5 times 2x plus 3