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Welcome to the video on completing the square.

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What's completing the square?

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Well, it's a way to solve a quadratic equation.

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And actually, let me just write down a quadratic equation, and

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then I will show you how to complete the square.

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And then we'll do another example, and then maybe talk

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a little bit about why it's called completing the square.

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So let's say I have this equation: x squared plus 16x

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minus 57 is equal to 0.

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So what are the tools in our toolkit right now that we

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could use to solve this?

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Well, we could try to factor it out.

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We could say, what two numbers add up to 16, and then when you

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multiply them they're minus 57?

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And you'd have to think about it a little bit.

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And you might get whole numbers, but you're not even

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sure if there are two whole numbers that work

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out like that.

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This problem there are.

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But, you know, sometimes the solution is a decimal number

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and you don't know it.

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So the only time you can really factor is if you're sure that

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you could factor this into kind of integer expressions.

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You know, x plus some integer or x minus some integer

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times, you know, x plus some other integer.

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Or likewise.

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The other option is to do the quadratic equation.

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And what we're going to see is actually the quadratic equation

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is just essentially a shortcut to completing the square.

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The quadratic equation is actually proven using

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completing the square.

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So what is completing the square?

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So what do we do?

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Well, before we move into this video let's see what happens

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if I square an expression.

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Let me do it in this down here.

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What is x plus a, squared?

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Well that equals x squared plus 2ax plus a squared.

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Right?

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So if you ever see something in this form, you know that it's

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x plus something squared.

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So wouldn't it be neat if we could manipulate this equation

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so we can write it as x plus a squared equals something,

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and then we could just take the square root?

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And what we're going to do is, actually, do just that.

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And that is completing the square.

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So let me show you an example.

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I think an example will make it a little clearer.

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Let me box this away.

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This is what you need to remember.

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This is the whole rationale behind competing the squares--

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to get an equation into this form, onto one side of the

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equation, and just have a number on the other side, so

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you could take the square root of both sides.

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So let's see.

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First of all, let's just check to make sure this isn't

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a perfect square.

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If this were, this coefficient would be equivalent to the 2a.

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Right?

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So a would be 8, and then this would be 64.

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This is clearly not 64, so this right here is not

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a squared expression.

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So what can we do?

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Well let me get rid of the 57 by adding 57 to both

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sides of this equation.

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So I would get x squared plus 16x is equal to 57.

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All I did is I added 57 to both sides of this equation.

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Now, what could I add here so that this, the left-hand side

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of this equation, becomes a square of some expression

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like x plus a?

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If you just follow this pattern down here, we have x squared

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plus 2ax-- so you could view this right here as 2ax.

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Right?

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That's 2ax.

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And then we need to add an a squared to it.

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Right?

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Plus a squared.

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And then we would have the form here.

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But we know from basic algebra that anything you do to one

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side of an equation you have to do to another.

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So we added an a squared here, so let's add an a

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squared here as well.

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And now you could essentially rewrite this as a square

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of some expression.

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But before that we have to figure out what a was?

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Well how do we do that?

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Well, what is a?

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If this expression right here is 2ax, what is a?

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Well 2a is going to equal 16, so a is equal to 8.

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And you could usually do that just by inspection;

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do it in your head.

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But if you wanted to see it done algebraically you could

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actually write 2ax is equal to 16x.

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And then divide both sides by 2x, and you get a is

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equal to 16x over 2x.

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And assuming that x doesn't equal 0 this evaluates to 8.

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So a is 8.

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So if a is 8 we could rewrite that expression-- I'll switch

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colors arbitrarily-- as x squared plus 16x

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plus a squared.

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Well, it's 64, because a is 8.

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Is equal to 57 plus 64.

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Right?

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I went through a fairly tedious explanation here, but all we've

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really done to get from there to there is we added 57 to both

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sides of this equation to kind of get it on the right-hand

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side, and then we added 64 to both sides of this equation.

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And why did I add 64 to both sides of this equation?

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So that the left-hand side expression takes this form.

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Now that the left-hand side expression takes this form

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I can rewrite it as what?

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x plus a, squared.

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I can rewrite it in this form.

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And we know that a is 8, so it becomes x plus 8, squared,

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is equal to-- and what's 57 plus 64?

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It's 121.

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Now we have what looks like a fairly straightforward-- it's

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still a quadratic equation, actually, because if you

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were to expand this side you'd get a quadratic.

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But we can solve this without using the quadratic equation

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or without having to factor.

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We can just take the square root of both sides of this.

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And if we take the square root of both sides what do we get?

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We get-- once again, arbitrarily switching colors--

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that x plus 8 is equal to, and remember this, the plus or

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minus square root of 121.

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And what's the square root of 121?

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Well it's 11, right?

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So then we come here.

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Let me box this away.

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This was just an aside.

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So we get x plus 8 is equal to plus or minus 11.

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And so x is equal to-- subtract 8 from both sides-- minus

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8 plus or minus 11.

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And so x could equal-- so minus 8 plus 11 is 3.

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Right?

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Let me make sure I did that right.

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x is equal to minus 8 plus or minus 11.

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Yes.

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That's right.

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So x could be equal to 3.

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And then if I took minus 8 minus 11, x could

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also equal minus 19.

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All right.

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And let's see if that makes sense.

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So in theory this should be able to be factored as x

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minus 3 times x plus 19 is equal to 0.

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Right?

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Because these are the two solutions of this equation.

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And that works out, right?

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Minus 3 times 19 is minus 57.

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And minus 3 plus 19 is plus 16x.

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We could have just immediately factored it this way, but if

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that wasn't obvious to us-- because, you know, at least

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19 is kind of a strange number-- we could do it

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completing the square.

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And so why is it called completing the square?

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Because you get it in this form and then you have to add this

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64 here to kind of complete the square-- to turn this

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left-hand expression into a squared expression.

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Let's do one more.

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And I'll do less explanation and more just chugging through

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the problem, and that actually might make it seem simpler.

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But this is going to be a hairier problem.

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So let's say I have 6x squared minus 7x minus 3 is equal to 0.

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You could try to factor it, but personally I don't

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enjoy factoring things when I have a coefficient.

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And you can say, oh well why don't we divide both sides

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of this equation by 6?

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But then you'd get a fraction here and a fraction here.

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And that's even worse to factor just by inspection.

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You could do the quadratic equation.

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And maybe I'll show you in a future video, the quadratic

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equation-- and I think I've already done one where I proved

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the quadratic equation.

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But the quadratic equation is essentially

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completing the square.

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It's kind of a shortcut.

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It's just kind of remembering the formula.

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But let's complete the square here, because that's what the

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point of this video was.

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So let's add the 3 to both sides of that equation.

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We could do-- well, let's add the 3 first.

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So you get 6 x squared minus 7x is equal to 3.

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I added 3 to both sides.

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And some teachers will leave the minus 3 here, and then try

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to figure out what to add to it and all of that.

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But I like to get it out of the way so that I can figure out

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very clearly what number I should put here.

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But I also don't like the 6 here.

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It just complicates things.

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I like to have it x plus a squared, not some square root

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coefficient on the x term.

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So let's divide both sides of this equation by 6, and you get

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x squared minus 7/6 x is equal to-- 3 divided by 6

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is equal to 1/2.

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And we could have made that our first step.

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We could have divided by 6 right at that first step.

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Anyway, now let's try to complete the square.

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So we have x squared-- I'm just going to open up some space--

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minus 7/6 x plus something is going to be equal to 1/2.

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And so we have to add something here so that this left-hand

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expression becomes a squared expression.

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So how do we do that?

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Well essentially we look at this coefficient, and keep

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in mind this is not just 7/6 it's minus 7/6.

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You take 1/2 of it, and then you square it.

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Right?

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Let me do it.

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x plus a, squared, is equal to x squared plus

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2ax plus a squared.

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Right?

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This is what you have to remember all the time.

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That's all completing the square is based off of.

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So what did I say just now?

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Well, this term is going to be 1/2 of this

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coefficient squared.

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And how do we know that?

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Because a is going to be 1/2 of this coefficient if you just

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do a little bit of pattern matching.

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So what's 1/2 of this coefficient?

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1/2 of minus 7/6 is minus 7/12.

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So if you want you could write a equals minus

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7/12 for our example.

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And I just multiplied this by 1/2.

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Right?

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So what do I add to both sides?

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I add a squared.

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So what's 7/12 squared?

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Well that's going to be 49/144.

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If I did it to the left-hand side I have to do it to

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the right-hand side.

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Plus 49/144.

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And now how can I simplify this left-hand side?

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What's our next step?

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Well we now know it is a perfect square.

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In fact, we know what a is. a is minus 7/12.

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And so we know that this left-hand side of this equation

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is x minus a-- or x plus a, but a is a negative number.

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So x plus a, and a is negative, squared.

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And if you want you can multiply this out and confirm

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that it truly equals this.

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And that is going to be equal to-- let's get a common

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denominator, 144.

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So 72 plus 49 equals 121.

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121/144.

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So we have x minus 7/12, all of that squared

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is equal to 121/144.

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So what do we do now?

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Well now we just take the square root of both

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sides of this equation.

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And I'm trying to free up some space.

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Switch to green.

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Let me partition this off.

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And we get x minus 7/12 is equal to the plus or minus

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square root of that.

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So plus or minus 11/12.

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Right?

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Square root of 121 is 11.

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Square root of 144 is 12.

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So then we could add 7/12 to both sides of this equation,

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and we get x is equal to 7/12 plus or minus 11/12.

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Well that equals 7 plus or minus 11/12.

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So what are the two options?

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7 plus 11 is 18, over 12.

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So x could equal 18/12, is 3/2.

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Or, what's 7 minus 11?

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That's minus 4/12.

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So it's minus 1/3.

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There you have it.

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That is completing the square.

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Hopefully you found that reasonably insightful.

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And if you want to prove the quadratic equation, all you

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have to do is instead of having numbers here, write a x squared

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plus bx plus c equals 0.

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And then complete the square using the a, b, and c's

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instead of numbers.

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And you will end up with the quadratic equation

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by this point.

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And I think I did that in a video.

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Let me know if I didn't and I'll do it for you.

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Anyway, I'll see you in the next video.