 Welcome back today. We are going to talk about completing the square This is a process that we use in mathematics for a number of different reasons today I'm just going to show you the process of completing the square and then later on I think in another video I will show you kind of where it is used how we can use Completing the square anyway to complete the square of Of a quadratic equation so when I say a quadratic equation It's an equation with an x squared in it and that is the x squared is the highest power That's how you know it's a quadratic equation. Okay, so to complete the square of x squared plus bx now notice here Remember that your standard form of your quadratic equation is a x squared Plus bx plus c that's this usually the standard form usually it's set equal to zero depending on what you want to do with it This is standard form number in front for a number in front here for b That's in front of the x and then a constant there sees what we usually call that now notice here at that x squared Plus bx that there's no a in front here. Well, that's actually kind of the point whenever you do completing the square You can only have a one in front here Remember there's a one here that we in that we know that is there We just don't write it because it's a little bit redundant because this is just a single x squared one x square So we don't write that one because it's just tedious to keep writing all those ones But notice the these examples that I have here all all down here out down the side with these x squared They're all ones same thing over here all of these are ones right here Okay, so whenever you do completing the square. That's one thing I got to remember is That your x squared part it's got to have a one in front of it on the other hand the b part right here It can be just about any number positive negative even odd. That doesn't really matter But anyway to complete the square what we do is we're going to add b divided by two Quantity squared, so we're going to take our b number divided by two and then square it This is going to give us a number so that we can factor our Quadratic equation, okay now at this point. It's a little bit confusing what we would be using for after you see a couple of examples You can see why okay, so here's the algebra behind it So what I want to do is I want to I want a number here so I can use to factor with okay, so Insight in this blank here what I'm going to do is I'm going to put b divided by two squares That's what that's what's going here and then with that number. I will be able to factor Okay, now a couple teachers who might say this differently factoring or bubbles or parentheses or or whatever they might use But this is called factoring these two bubbles are the two different factors And then notice that these two factors can condense down into a single parentheses Okay, now looking at the algebra portion of this again, maybe a little bit confusing depending on your level of Understanding of this so that's why I have this number example over here Okay, so what if I have x squared minus 8x plus something okay? That's something I want to try to fill in I want to try to find a number there okay, so in this case I take The b number divided by 2 and then I square it well in this case my b number is negative 8 So negative 8 divided by 2 is negative 4 negative 4 when you square it is going to give you 16 a positive 16 okay now That number is is going to work out perfectly for us because that's 16 What that's going to be able to do is that's going to be able to give us the ability to Factor to go from this trinomial to two binomials is what we have down here Okay, x minus 4 x minus 4 the negative 4 negative 4 they multiply back to 16 And then these two are going to add to get negative a that's kind of the shortcut that you use for when you try to factor Okay, now from there since these both these of parentheses are the same There's both these parentheses are the same I can condense them down to a single parentheses squared I usually like to call this a single quantity squared okay now. I've kind of gone over the process Let's try a couple of examples just a few examples just to kind of hash out what what it means by completing the square What what do you mean find this number to use the factor? Okay, so just a quick couple of examples and then we'll finish up the video okay complete the square for each one of These expressions I got two of them here write the resulting expression as a binomial squared So just the exact same thing I was doing on the previous slide We're just now going to do this with just a couple of different examples All right on the first one here The first thing I want to do is find out what number to plug in here. So that's where I do my b over 2 Quantity squared then I'm going to take this b number in this case. It's a negative 14 So I got to take negative 14 Divided by 2 and I'm going to square it so in that case. I don't have too much room here So negative 14 squared or see me negative 14 divided by 2 is negative 7 Negative 7 squared is going to be a positive 49 now That's going to happen every time you're always going to get positive numbers when you do this Of course as you're squaring 2 negative is going to make a positive that makes sense. So here we go x squared minus 14x Plus 49 there we are so that's the number that's going to give me the ability to factor That's the that's the that's the reason we use completing the square is so that we can find this number So we can factor okay, so I'm going to get my parentheses here Okay, find my two factors. I got x's in the front Two numbers that multiply to 49 and add to negative 14 is going to be a negative 7 and a negative 7 Okay, they both got to be negative to get this positive here And then they both have to be negative to get a negative 14 Okay, so that kind of makes sense how they set that up okay So this portion right here where my mouse is that get that right there where my cursor is that this is completing the square Finding that number that is completing the square The rest of this is kind of all right factoring and condensing and that kind of stuff Okay, so now what I'm going to do is just as it says here right through resulting expression as a binomial squared So here's my binomial x minus seven squared as You do more and more of these in your math class you'll realize that this is going to happen every single time I want you to get the same number in these parentheses. So I'm always going to be able to condense it down Okay, you can't always do that with every parentheses But in this case you can actually do that because they're the exact same on the left and the right It's just really really nice. Okay, so to this next example over here now You might notice some of you might notice ahead of time. We got a nine here. I'm going to divide that by two I'm going to take that b number divided by two which causes a bit a trouble Okay, but we're going to work our way through this. All right. So anyway, just keep going through the process I need to take this Be divided by two. I can need to find out what this number is right here. I got to find out how to complete the square Okay, so in this case That b number is nine divided by two squared. Now, this is where I'm going to suggest that you don't use decimals. Do not take nine divided by two and get four point five Don't do that. Okay, instead just square the top and square the bottom Get 81 over four Now yes, it looks like a top-heavy improper fraction all that kind of jazz, but it's still It's more useful than a decimal In this case, I think we understand fractions here a little bit better than decimals You'll see here when I'm talking about here a little bit once we go through this example. Okay, so now that's the number I add 9x plus 81 over 4 add inside this little parentheses Now this will give me the ability to factor and then you're telling then you're asking yourself well that 81 over 4 is not going to be nice because What numbers multiply to 81 over 4 and it add to 9 that just doesn't make any sense. Yes, our Concept of what those numbers are with mixing fraction fractions and numbers together like the fractions here and numbers here together We really don't have a great understanding of that at this level, but what we can do I'm going to flip back a page here is That if we follow this process if you follow this algebra process here notice, okay? If I take any number, okay? Take that number divided by two squared whatever that number might be when I get to this point I get from here to here to factor notice this number we can figure out what it is I don't have to I don't have to figure out what numbers multiply to the end and add to the middle I don't really need to do that all I did number that I need right here is just be divided by two be divided by two That's all I need. Okay, so to go back to our problem Okay, the only number that I need right here is just be divided by two and Be divided by two so in this case nine divided by two and nine divided by two so there I didn't really need to figure out. Okay, what numbers multiply to 81 over 4 and add to 9 I didn't really need to do that because I looked at my notes and I followed the process Okay, now after you put in the numbers now you might be able to see it So okay 9 times 9 is 81 2 times 2 is 4 and then if you add these together This is 4 4 and a half 4 and a half which adds together to get 9 So now you might be able to actually see that but before seeing that answer You might not anyway last step condense this down x plus 9 halves squared Okay, so there's my binomial squared All right So that was just a quick video on how to complete the square and just a little bit of a couple of question a couple of Different examples of how to use it and little things like that I will do another video another video of how to solve how to use Completing the square to solve quadratic equations and some other uses for it But I hope hope you enjoyed the video. Hope you learned something today and we'll see you next time