Quick Guide to Completing the Square

What's up guys in this video I want to give you a quick guide to completing the square. Now there's a lot of reasons that we want to complete the square. We can use that to write it in vertex form to identify the vertex of the parabola and sometimes we can use completing the square to solve them. But in this example what I want to do is just kind of like give you the basic idea of what we're trying to achieve when we are completing the square. And I think the main thing that I want you to understand is the idea of creating this perfect square. So let me just go ahead and change this out a little bit and I think this is a rule that I want you to use anytime that you are struggling. One thing I always like to tell my students like you know when you're learning something you're taking a test or anything else and maybe the numbers kind of seem tricky or you're not really sure what to do like go back and use some basic numbers okay. So what I'm going to do is I'm going to actually change this equation over here to something that's a little bit easier for me to understand. All right now you might be wondering well why did you pick these numbers like what's special about this? Well the reason what's special about this is this is what we call a perfect square trinomial. Now remember perfect square trinomials are are going to be trinomials that your first term is squared and your last term is squared. And what's so important about a perfect square trinomial is that they can be factored down into a binomial squared. So you can see like in this example here this is exactly what I'm looking for. This is going to be my vertex form. That's what I'm trying to create when I am completing the square. One thing that I you know I want you to kind of idea is like when you're looking at you know trying to complete the square I'm going to kind of go through this in two different methods on two different ways. One thing I want you to think about is like all right if I need to take this and I need to create this binomial squared right that's what is putting things into vertex form like what number do I need? Well obviously I already have an x squared and a 6x right. I just need that 9. I don't want the 8. I just needed that 9. So one thing you could do is just say well all right well why don't we just add the 9 then right. And let's just go ahead and add the 9. I'll make that in yellow here and then obviously we know that well you just can't like add a 9 to one side right. Remember we like either going to add 9 to one side you have to add 9 to the other side all right. And then to remember like the other thing is like don't forget about the 8. Like the 8 is still there you just can't like not forget about the 8. So let's just go ahead and write the 8 over here okay. Now here's something that's really important actually let me write this in with okay. So now what we did is we knew if we had an x squared plus a 6x we knew the number that was going to create that perfect square trinomial was a 9. So I just added that 9 in there right. Kind of like some math magic but it's okay because I added a 9 onto both sides right. I mean again if you have an equation as long as you're doing one thing to one side you're doing the same thing to the other side it's going to be okay. You're still keeping an equivalent equation. So now I know that this factors down to this binomial squared. So now I can have a let's see an x plus 3 quantity squared. Now again if I want to solve for y right what are you going to do? Well you're going to subtract a 9 on both sides and therefore remember this is going to there so 8 minus 9 is going to be a minus 1 and there you go. Now there is a way to kind of work through a problem when you kind of know what you're working with as far as creating a perfect square trinomial. But let me go through this exact same process if we didn't already recognize a perfect square trinomial. If we didn't know x squared plus 6x that is going to give us 9. So let me just kind of do the exact same problem but also give you some some different ways to kind of approach it. So one thing that you always want to do is we have these first two terms. So again that is what you're going to want to look into creating that perfect square trinomial. So what we'll do is we'll put parentheses around them. Then the next thing we want to do is like this one was obvious right I kind of like like I wanted you to recognize that relationship. But one of the hard things about completing the square is not always as obvious what the what the value is that creates the perfect square trinomial. So what we can do to always find that value is take our b divided by 2 and square it. So in this case you would have a 6 divided by 2 squared which is going to equal a 9. Now rather than adding a 9 to both sides notice what I had to do. I just subtract it right from both sides to solve for the y. So what I like to do is I like to add and subtract it on the same side. So I have y is equal to a x squared plus a 6x right. We're going to do a plus 9 and then we're gonna do is a minus 9 and then plus 8 right. Because isn't that exactly what happened anyways like I had to add the 9 then I had to subtract the 9 anyways. So this is just a this is just making this process go by much quicker. And now I have created my perfect square trinomial so I can go ahead and factor it down. So that is a quick easy way to complete the square but if you want to know a step-by-step method for completing the square then that's going to come up in the next video.