0:00:00.870,0:00:03.610
Welcome to the video on[br]completing the square.

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What's completing the square?

0:00:04.440,0:00:06.740
Well, it's a way to solve[br]a quadratic equation.

0:00:06.740,0:00:09.700
And so before I--actually, let me just write[br]down a quadratic equation, and

0:00:09.700,0:00:11.570
then I will show you how[br]to complete the square.

0:00:11.570,0:00:13.460
And then we'll do another[br]example, and then maybe we'll talk

0:00:13.460,0:00:16.650
a little bit about why it's[br]called completing the square.

0:00:16.650,0:00:27.770
So let's say I have this[br]equation: x squared plus 16x

0:00:27.770,0:00:32.600
minus 57 is equal to 0.

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

0:00:36.130,0:00:36.970
could use to solve this?

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Well, we could try[br]to factor it out.

0:00:38.570,0:00:41.770
We could say, what two numbers[br]add up to 16, and then when you

0:00:41.770,0:00:44.060
multiply them they're minus 57?

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

0:00:45.450,0:00:47.360
And you might get whole[br]numbers, but you're not even

0:00:47.360,0:00:49.050
sure if there are two whole[br]numbers that work

0:00:49.050,0:00:49.540
out like that.

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This problem there are.

0:00:50.630,0:00:53.510
But, you know, sometimes the[br]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[br]factor is if you're sure that

0:00:58.150,0:01:01.000
you could factor this into[br]kind of integer expressions.

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You know, x plus some integer[br]or x minus some integer

0:01:03.620,0:01:05.920
times, you know, x plus[br]some other integer.

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Or likewise.

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The other option is to do[br]the quadratic equation.

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

0:01:11.420,0:01:15.510
is just essentially a shortcut[br]to completing the square.

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The quadratic equation is[br]actually proven using

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completing the square.

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So what is completing[br]the square?

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So what do we do?

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Well, before we move into this[br]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[br]plus 2ax plus a squared.

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Right?

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

0:01:55.420,0:01:57.740
x plus something squared.

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

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

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and then we could just[br]take the square root?

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

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And that is completing[br]the square.

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So let me show you an example.

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I think an example will[br]make it a little clearer.

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Let me box this away.

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This is what you[br]need to remember.

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This is the whole rationale[br]behind competing the squares--

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

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

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you could take the square[br]root of both sides.

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So let's see.

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

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a perfect square.

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

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Right?

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So a would be 8, and[br]then this would be 64.

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This is clearly not 64, so[br]this right here is not

0:02:48.270,0:02:50.840
a squared expression.

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So what can we do?

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

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sides of this equation.

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

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

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

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of this equation, becomes a[br]square of some expression

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like x plus a?

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

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plus 2ax-- so you could view[br]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[br]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[br]have the form here.

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

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side of an equation you[br]have to do to another.

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

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squared here as well.

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And now you could essentially[br]rewrite this as a square

0:04:01.350,0:04:02.260
of some expression.

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But before that we have to[br]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[br]here is 2ax, what is a?

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

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And you could usually do[br]that just by inspection;

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do it in your head.

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But if you wanted to see it[br]done algebraically you could

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actually write 2ax[br]is equal to 16x.

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And then divide both sides[br]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[br]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[br]that expression-- I'll switch

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colors arbitrarily-- as x[br]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?

0:05:00.720,0:05:04.600
I went through a fairly tedious[br]explanation here, but all we've

0:05:04.600,0:05:08.890
really done to get from there[br]to there is we added 57 to both

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

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

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

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

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Now that the left-hand side[br]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[br]becomes x plus 8, squared,

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

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It's 121.

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

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still a quadratic equation,[br]actually, because if you

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

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But we can solve this without[br]using the quadratic equation

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or without having to factor.

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

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

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We get-- once again,[br]arbitrarily switching colors--

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

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minus square root of 121.

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And what's the[br]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[br]to plus or minus 11.

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

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8 plus or minus 11.

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

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Right?

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Let me make sure I[br]did that right.

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x is equal to minus[br]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[br]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[br]that makes sense.

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So in theory this should be[br]able to be factored as x

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minus 3 times x plus[br]19 is equal to 0.

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Right?

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Because these are the two[br]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[br]19 is plus 16x.

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

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

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

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completing the square.

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And so why is it called[br]completing the square?

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

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64 here to kind of complete the[br]square-- to turn this

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left-hand expression into[br]a squared expression.

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Let's do one more.

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

0:07:59.920,0:08:02.105
the problem, and that actually[br]might make it seem simpler.

0:08:04.800,0:08:07.080
But this is going to[br]be a hairier problem.

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

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

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enjoy factoring things[br]when I have a coefficient.

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And you can say, oh well why[br]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[br]here and a fraction here.

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

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You could do the[br]quadratic equation.

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

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

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the quadratic equation.

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But the quadratic[br]equation is essentially

0:08:42.380,0:08:43.170
completing the square.

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It's kind of a shortcut.

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It's just kind of[br]remembering the formula.

0:08:46.280,0:08:48.320
But let's complete the square[br]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[br]sides of that equation.

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

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

0:09:04.820,0:09:06.770
I added 3 to both sides.

0:09:06.770,0:09:09.470
And some teachers will leave[br]the minus 3 here, and then try

0:09:09.470,0:09:11.050
to figure out what to add[br]to it and all of that.

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

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very clearly what number[br]I should put here.

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But I also don't[br]like the 6 here.

0:09:18.230,0:09:19.550
It just complicates things.

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

0:09:25.990,0:09:27.450
coefficient on the x term.

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

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

0:09:39.730,0:09:41.566
is equal to 1/2.

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And we could have made[br]that our first step.

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

0:09:46.450,0:09:49.250
Anyway, now let's try to[br]complete the square.

0:09:49.250,0:09:51.800
So we have x squared-- I'm just[br]going to open up some space--

0:09:51.800,0:09:59.530
minus 7/6 x plus something is[br]going to be equal to 1/2.

0:09:59.530,0:10:02.400
And so we have to add something[br]here so that this left-hand

0:10:02.400,0:10:05.290
expression becomes a[br]squared expression.

0:10:05.290,0:10:06.620
So how do we do that?

0:10:06.620,0:10:10.770
Well essentially we look at[br]this coefficient, and keep

0:10:10.770,0:10:14.610
in mind this is not just[br]7/6 it's minus 7/6.

0:10:14.610,0:10:17.460
You take 1/2 of it, and[br]then you square it.

0:10:17.460,0:10:18.610
Right?

0:10:18.610,0:10:19.690
Let me do it.

0:10:19.690,0:10:25.290
x plus a, squared, is[br]equal to x squared plus

0:10:25.290,0:10:28.820
2ax plus a squared.

0:10:28.820,0:10:29.070
Right?

0:10:29.070,0:10:30.750
This is what you have to[br]remember all the time.

0:10:30.750,0:10:33.560
That's all completing the[br]square is based off of.

0:10:33.560,0:10:34.980
So what did I say just now?

0:10:34.980,0:10:37.260
Well, this term is going[br]to be 1/2 of this

0:10:37.260,0:10:39.190
coefficient squared.

0:10:39.190,0:10:40.190
And how do we know that?

0:10:40.190,0:10:43.880
Because a is going to be 1/2 of[br]this coefficient if you just

0:10:43.880,0:10:45.850
do a little bit of[br]pattern matching.

0:10:45.850,0:10:48.760
So what's 1/2 of[br]this coefficient?

0:10:48.760,0:10:54.050
1/2 of minus 7/6 is minus 7/12.

0:10:54.050,0:10:56.640
So if you want you could[br]write a equals minus

0:10:56.640,0:10:58.770
7/12 for our example.

0:10:58.770,0:11:00.770
And I just multiplied[br]this by 1/2.

0:11:00.770,0:11:01.980
Right?

0:11:01.980,0:11:03.660
So what do I add to both sides?

0:11:03.660,0:11:06.030
I add a squared.

0:11:06.030,0:11:08.930
So what's 7/12 squared?

0:11:08.930,0:11:13.220
Well that's going to be 49/144.

0:11:13.220,0:11:15.000
If I did it to the left-hand[br]side I have to do it to

0:11:15.000,0:11:16.630
the right-hand side.

0:11:16.630,0:11:22.240
Plus 49/144.

0:11:22.240,0:11:26.120
And now how can I simplify[br]this left-hand side?

0:11:26.120,0:11:26.880
What's our next step?

0:11:26.880,0:11:28.470
Well we now know it[br]is a perfect square.

0:11:28.470,0:11:31.550
In fact, we know what a[br]is. a is minus 7/12.

0:11:31.550,0:11:35.200
And so we know that this[br]left-hand side of this equation

0:11:35.200,0:11:43.390
is x minus a-- or x plus a,[br]but a is a negative number.

0:11:43.390,0:11:47.980
So x plus a, and a is[br]negative, squared.

0:11:47.980,0:11:50.350
And if you want you can[br]multiply this out and confirm

0:11:50.350,0:11:53.130
that it truly equals this.

0:11:53.130,0:11:55.920
And that is going to be equal[br]to-- let's get a common

0:11:55.920,0:11:58.360
denominator, 144.

0:11:58.360,0:12:04.070
So 72 plus 49 equals 121.

0:12:04.070,0:12:06.300
121/144.

0:12:06.300,0:12:09.210
So we have x minus 7/12,[br]all of that squared

0:12:09.210,0:12:13.180
is equal to 121/144.

0:12:13.180,0:12:14.300
So what do we do now?

0:12:14.300,0:12:15.570
Well now we just take[br]the square root of both

0:12:15.570,0:12:17.700
sides of this equation.

0:12:17.700,0:12:20.140
And I'm trying to[br]free up some space.

0:12:20.140,0:12:22.215
Switch to green.

0:12:22.215,0:12:25.320
Let me partition this off.

0:12:25.320,0:12:33.310
And we get x minus 7/12 is[br]equal to the plus or minus

0:12:33.310,0:12:33.940
square root of that.

0:12:33.940,0:12:38.120
So plus or minus 11/12.

0:12:38.120,0:12:38.390
Right?

0:12:38.390,0:12:39.660
Square root of 121 is 11.

0:12:39.660,0:12:42.420
Square root of 144 is 12.

0:12:42.420,0:12:44.480
So then we could add 7/12 to[br]both sides of this equation,

0:12:44.480,0:12:53.100
and we get x is equal to[br]7/12 plus or minus 11/12.

0:12:53.100,0:12:58.660
Well that equals 7[br]plus or minus 11/12.

0:12:58.660,0:13:00.050
So what are the two options?

0:13:00.050,0:13:03.930
7 plus 11 is 18, over 12.

0:13:03.930,0:13:08.210
So x could equal 18/12, is 3/2.

0:13:08.210,0:13:11.010
Or, what's 7 minus 11?

0:13:11.010,0:13:12.760
That's minus 4/12.

0:13:12.760,0:13:15.370
So it's minus 1/3.

0:13:15.370,0:13:16.630
There you have it.

0:13:16.630,0:13:17.940
That is completing the square.

0:13:17.940,0:13:20.220
Hopefully you found that[br]reasonably insightful.

0:13:20.220,0:13:23.340
And if you want to prove the[br]quadratic equation, all you

0:13:23.340,0:13:27.320
have to do is instead of having[br]numbers here, write a x squared

0:13:27.320,0:13:29.820
plus bx plus c equals 0.

0:13:29.820,0:13:34.130
And then complete the square[br]using the a, b, and c's

0:13:34.130,0:13:35.060
instead of numbers.

0:13:35.060,0:13:37.180
And you will end up with[br]the quadratic equation

0:13:37.180,0:13:38.110
by this point.

0:13:38.110,0:13:39.510
And I think I did[br]that in a video.

0:13:39.510,0:13:41.600
Let me know if I didn't[br]and I'll do it for you.

0:13:41.600,0:13:44.540
Anyway, I'll see you[br]in the next video.