[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.87,0:00:03.61,Default,,0000,0000,0000,,Welcome to the video on\Ncompleting the square.
Dialogue: 0,0:00:03.61,0:00:04.44,Default,,0000,0000,0000,,What's completing the square?
Dialogue: 0,0:00:04.44,0:00:06.74,Default,,0000,0000,0000,,Well, it's a way to solve\Na quadratic equation.
Dialogue: 0,0:00:06.74,0:00:09.70,Default,,0000,0000,0000,,And so before I--actually, let me just write\Ndown a quadratic equation, and
Dialogue: 0,0:00:09.70,0:00:11.57,Default,,0000,0000,0000,,then I will show you how\Nto complete the square.
Dialogue: 0,0:00:11.57,0:00:13.46,Default,,0000,0000,0000,,And then we'll do another\Nexample, and then maybe we'll talk
Dialogue: 0,0:00:13.46,0:00:16.65,Default,,0000,0000,0000,,a little bit about why it's\Ncalled completing the square.
Dialogue: 0,0:00:16.65,0:00:27.77,Default,,0000,0000,0000,,So let's say I have this\Nequation: x squared plus 16x
Dialogue: 0,0:00:27.77,0:00:32.60,Default,,0000,0000,0000,,minus 57 is equal to 0.
Dialogue: 0,0:00:32.60,0:00:36.13,Default,,0000,0000,0000,,So what are the tools in our\Ntoolkit right now that we
Dialogue: 0,0:00:36.13,0:00:36.97,Default,,0000,0000,0000,,could use to solve this?
Dialogue: 0,0:00:36.97,0:00:38.57,Default,,0000,0000,0000,,Well, we could try\Nto factor it out.
Dialogue: 0,0:00:38.57,0:00:41.77,Default,,0000,0000,0000,,We could say, what two numbers\Nadd up to 16, and then when you
Dialogue: 0,0:00:41.77,0:00:44.06,Default,,0000,0000,0000,,multiply them they're minus 57?
Dialogue: 0,0:00:44.06,0:00:45.45,Default,,0000,0000,0000,,And you'd have to think\Nabout it a little bit.
Dialogue: 0,0:00:45.45,0:00:47.36,Default,,0000,0000,0000,,And you might get whole\Nnumbers, but you're not even
Dialogue: 0,0:00:47.36,0:00:49.05,Default,,0000,0000,0000,,sure if there are two whole\Nnumbers that work
Dialogue: 0,0:00:49.05,0:00:49.54,Default,,0000,0000,0000,,out like that.
Dialogue: 0,0:00:49.54,0:00:50.63,Default,,0000,0000,0000,,This problem there are.
Dialogue: 0,0:00:50.63,0:00:53.51,Default,,0000,0000,0000,,But, you know, sometimes the\Nsolution is a decimal number
Dialogue: 0,0:00:53.51,0:00:54.19,Default,,0000,0000,0000,,and you don't know it.
Dialogue: 0,0:00:54.19,0:00:58.15,Default,,0000,0000,0000,,So the only time you can really\Nfactor is if you're sure that
Dialogue: 0,0:00:58.15,0:01:01.00,Default,,0000,0000,0000,,you could factor this into\Nkind of integer expressions.
Dialogue: 0,0:01:01.00,0:01:03.62,Default,,0000,0000,0000,,You know, x plus some integer\Nor x minus some integer
Dialogue: 0,0:01:03.62,0:01:05.92,Default,,0000,0000,0000,,times, you know, x plus\Nsome other integer.
Dialogue: 0,0:01:05.92,0:01:06.99,Default,,0000,0000,0000,,Or likewise.
Dialogue: 0,0:01:06.99,0:01:09.24,Default,,0000,0000,0000,,The other option is to do\Nthe quadratic equation.
Dialogue: 0,0:01:09.24,0:01:11.42,Default,,0000,0000,0000,,And what we're going to see is\Nactually the quadratic equation
Dialogue: 0,0:01:11.42,0:01:15.51,Default,,0000,0000,0000,,is just essentially a shortcut\Nto completing the square.
Dialogue: 0,0:01:15.51,0:01:18.41,Default,,0000,0000,0000,,The quadratic equation is\Nactually proven using
Dialogue: 0,0:01:18.41,0:01:19.42,Default,,0000,0000,0000,,completing the square.
Dialogue: 0,0:01:19.42,0:01:21.42,Default,,0000,0000,0000,,So what is completing\Nthe square?
Dialogue: 0,0:01:21.42,0:01:23.34,Default,,0000,0000,0000,,So what do we do?
Dialogue: 0,0:01:23.34,0:01:27.08,Default,,0000,0000,0000,,Well, before we move into this\Nvideo let's see what happens
Dialogue: 0,0:01:27.08,0:01:30.93,Default,,0000,0000,0000,,if I square an expression.
Dialogue: 0,0:01:30.93,0:01:33.22,Default,,0000,0000,0000,,Let me do it in this down here.
Dialogue: 0,0:01:33.22,0:01:40.25,Default,,0000,0000,0000,,What is x plus a, squared?
Dialogue: 0,0:01:40.25,0:01:50.94,Default,,0000,0000,0000,,Well that equals x squared\Nplus 2ax plus a squared.
Dialogue: 0,0:01:50.94,0:01:51.68,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:01:51.68,0:01:55.42,Default,,0000,0000,0000,,So if you ever see something in\Nthis form, you know that it's
Dialogue: 0,0:01:55.42,0:01:57.74,Default,,0000,0000,0000,,x plus something squared.
Dialogue: 0,0:01:57.74,0:02:01.04,Default,,0000,0000,0000,,So wouldn't it be neat if we\Ncould manipulate this equation
Dialogue: 0,0:02:01.04,0:02:05.90,Default,,0000,0000,0000,,so we can write it as x plus a\Nsquared equals something,
Dialogue: 0,0:02:05.90,0:02:08.14,Default,,0000,0000,0000,,and then we could just\Ntake the square root?
Dialogue: 0,0:02:08.14,0:02:11.58,Default,,0000,0000,0000,,And what we're going to do\Nis, actually, do just that.
Dialogue: 0,0:02:11.58,0:02:13.09,Default,,0000,0000,0000,,And that is completing\Nthe square.
Dialogue: 0,0:02:13.09,0:02:15.01,Default,,0000,0000,0000,,So let me show you an example.
Dialogue: 0,0:02:15.01,0:02:16.52,Default,,0000,0000,0000,,I think an example will\Nmake it a little clearer.
Dialogue: 0,0:02:16.52,0:02:17.62,Default,,0000,0000,0000,,Let me box this away.
Dialogue: 0,0:02:17.62,0:02:19.31,Default,,0000,0000,0000,,This is what you\Nneed to remember.
Dialogue: 0,0:02:19.31,0:02:22.13,Default,,0000,0000,0000,,This is the whole rationale\Nbehind competing the squares--
Dialogue: 0,0:02:22.13,0:02:25.65,Default,,0000,0000,0000,,to get an equation into this\Nform, onto one side of the
Dialogue: 0,0:02:25.65,0:02:27.94,Default,,0000,0000,0000,,equation, and just have a\Nnumber on the other side, so
Dialogue: 0,0:02:27.94,0:02:31.21,Default,,0000,0000,0000,,you could take the square\Nroot of both sides.
Dialogue: 0,0:02:31.21,0:02:32.00,Default,,0000,0000,0000,,So let's see.
Dialogue: 0,0:02:32.00,0:02:33.97,Default,,0000,0000,0000,,First of all, let's just check\Nto make sure this isn't
Dialogue: 0,0:02:33.97,0:02:35.02,Default,,0000,0000,0000,,a perfect square.
Dialogue: 0,0:02:35.02,0:02:39.70,Default,,0000,0000,0000,,If this were, this coefficient\Nwould be equivalent to the 2a.
Dialogue: 0,0:02:39.70,0:02:40.47,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:02:40.47,0:02:44.44,Default,,0000,0000,0000,,So a would be 8, and\Nthen this would be 64.
Dialogue: 0,0:02:44.44,0:02:48.27,Default,,0000,0000,0000,,This is clearly not 64, so\Nthis right here is not
Dialogue: 0,0:02:48.27,0:02:50.84,Default,,0000,0000,0000,,a squared expression.
Dialogue: 0,0:02:50.84,0:02:51.68,Default,,0000,0000,0000,,So what can we do?
Dialogue: 0,0:02:51.68,0:02:55.99,Default,,0000,0000,0000,,Well let me get rid of the\N57 by adding 57 to both
Dialogue: 0,0:02:55.99,0:02:57.20,Default,,0000,0000,0000,,sides of this equation.
Dialogue: 0,0:02:57.20,0:03:07.55,Default,,0000,0000,0000,,So I would get x squared\Nplus 16x is equal to 57.
Dialogue: 0,0:03:07.55,0:03:11.47,Default,,0000,0000,0000,,All I did is I added 57 to\Nboth sides of this equation.
Dialogue: 0,0:03:11.47,0:03:16.30,Default,,0000,0000,0000,,Now, what could I add here so\Nthat this, the left-hand side
Dialogue: 0,0:03:16.30,0:03:21.48,Default,,0000,0000,0000,,of this equation, becomes a\Nsquare of some expression
Dialogue: 0,0:03:21.48,0:03:24.82,Default,,0000,0000,0000,,like x plus a?
Dialogue: 0,0:03:24.82,0:03:28.79,Default,,0000,0000,0000,,If you just follow this pattern\Ndown here, we have x squared
Dialogue: 0,0:03:28.79,0:03:37.88,Default,,0000,0000,0000,,plus 2ax-- so you could view\Nthis right here as 2ax.
Dialogue: 0,0:03:37.88,0:03:39.09,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:39.09,0:03:40.90,Default,,0000,0000,0000,,That's 2ax.
Dialogue: 0,0:03:40.90,0:03:43.52,Default,,0000,0000,0000,,And then we need to add\Nan a squared to it.
Dialogue: 0,0:03:43.52,0:03:44.04,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:44.04,0:03:46.30,Default,,0000,0000,0000,,Plus a squared.
Dialogue: 0,0:03:46.30,0:03:48.01,Default,,0000,0000,0000,,And then we would\Nhave the form here.
Dialogue: 0,0:03:48.01,0:03:50.51,Default,,0000,0000,0000,,But we know from basic algebra\Nthat anything you do to one
Dialogue: 0,0:03:50.51,0:03:52.08,Default,,0000,0000,0000,,side of an equation you\Nhave to do to another.
Dialogue: 0,0:03:52.08,0:03:54.23,Default,,0000,0000,0000,,So we added an a squared\Nhere, so let's add an a
Dialogue: 0,0:03:54.23,0:03:56.84,Default,,0000,0000,0000,,squared here as well.
Dialogue: 0,0:03:56.84,0:04:01.35,Default,,0000,0000,0000,,And now you could essentially\Nrewrite this as a square
Dialogue: 0,0:04:01.35,0:04:02.26,Default,,0000,0000,0000,,of some expression.
Dialogue: 0,0:04:02.26,0:04:04.21,Default,,0000,0000,0000,,But before that we have to\Nfigure out what a was?
Dialogue: 0,0:04:04.21,0:04:05.52,Default,,0000,0000,0000,,Well how do we do that?
Dialogue: 0,0:04:05.52,0:04:06.74,Default,,0000,0000,0000,,Well, what is a?
Dialogue: 0,0:04:06.74,0:04:10.72,Default,,0000,0000,0000,,If this expression right\Nhere is 2ax, what is a?
Dialogue: 0,0:04:10.72,0:04:15.38,Default,,0000,0000,0000,,Well 2a is going to equal\N16, so a is equal to 8.
Dialogue: 0,0:04:15.38,0:04:18.02,Default,,0000,0000,0000,,And you could usually do\Nthat just by inspection;
Dialogue: 0,0:04:18.02,0:04:18.63,Default,,0000,0000,0000,,do it in your head.
Dialogue: 0,0:04:18.63,0:04:20.93,Default,,0000,0000,0000,,But if you wanted to see it\Ndone algebraically you could
Dialogue: 0,0:04:20.93,0:04:25.69,Default,,0000,0000,0000,,actually write 2ax\Nis equal to 16x.
Dialogue: 0,0:04:25.69,0:04:29.09,Default,,0000,0000,0000,,And then divide both sides\Nby 2x, and you get a is
Dialogue: 0,0:04:29.09,0:04:31.43,Default,,0000,0000,0000,,equal to 16x over 2x.
Dialogue: 0,0:04:31.43,0:04:36.95,Default,,0000,0000,0000,,And assuming that x doesn't\Nequal 0 this evaluates to 8.
Dialogue: 0,0:04:36.95,0:04:38.13,Default,,0000,0000,0000,,So a is 8.
Dialogue: 0,0:04:38.13,0:04:42.43,Default,,0000,0000,0000,,So if a is 8 we could rewrite\Nthat expression-- I'll switch
Dialogue: 0,0:04:42.43,0:04:49.03,Default,,0000,0000,0000,,colors arbitrarily-- as x\Nsquared plus 16x
Dialogue: 0,0:04:49.03,0:04:50.47,Default,,0000,0000,0000,,plus a squared.
Dialogue: 0,0:04:50.47,0:04:54.18,Default,,0000,0000,0000,,Well, it's 64, because a is 8.
Dialogue: 0,0:04:54.18,0:04:59.17,Default,,0000,0000,0000,,Is equal to 57 plus 64.
Dialogue: 0,0:04:59.17,0:05:00.72,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:05:00.72,0:05:04.60,Default,,0000,0000,0000,,I went through a fairly tedious\Nexplanation here, but all we've
Dialogue: 0,0:05:04.60,0:05:08.89,Default,,0000,0000,0000,,really done to get from there\Nto there is we added 57 to both
Dialogue: 0,0:05:08.89,0:05:10.87,Default,,0000,0000,0000,,sides of this equation to kind\Nof get it on the right-hand
Dialogue: 0,0:05:10.87,0:05:14.32,Default,,0000,0000,0000,,side, and then we added 64 to\Nboth sides of this equation.
Dialogue: 0,0:05:14.32,0:05:16.83,Default,,0000,0000,0000,,And why did I add 64 to both\Nsides of this equation?
Dialogue: 0,0:05:16.83,0:05:21.07,Default,,0000,0000,0000,,So that the left-hand side\Nexpression takes this form.
Dialogue: 0,0:05:21.07,0:05:23.20,Default,,0000,0000,0000,,Now that the left-hand side\Nexpression takes this form
Dialogue: 0,0:05:23.20,0:05:26.03,Default,,0000,0000,0000,,I can rewrite it as what?
Dialogue: 0,0:05:26.03,0:05:27.17,Default,,0000,0000,0000,,x plus a, squared.
Dialogue: 0,0:05:27.17,0:05:28.62,Default,,0000,0000,0000,,I can rewrite it in this form.
Dialogue: 0,0:05:28.62,0:05:35.55,Default,,0000,0000,0000,,And we know that a is 8, so it\Nbecomes x plus 8, squared,
Dialogue: 0,0:05:35.55,0:05:39.73,Default,,0000,0000,0000,,is equal to-- and\Nwhat's 57 plus 64?
Dialogue: 0,0:05:39.73,0:05:43.09,Default,,0000,0000,0000,,It's 121.
Dialogue: 0,0:05:43.09,0:05:47.27,Default,,0000,0000,0000,,Now we have what looks like a\Nfairly straightforward-- it's
Dialogue: 0,0:05:47.27,0:05:48.96,Default,,0000,0000,0000,,still a quadratic equation,\Nactually, because if you
Dialogue: 0,0:05:48.96,0:05:50.35,Default,,0000,0000,0000,,were to expand this side\Nyou'd get a quadratic.
Dialogue: 0,0:05:50.35,0:05:53.06,Default,,0000,0000,0000,,But we can solve this without\Nusing the quadratic equation
Dialogue: 0,0:05:53.06,0:05:54.61,Default,,0000,0000,0000,,or without having to factor.
Dialogue: 0,0:05:54.61,0:05:57.39,Default,,0000,0000,0000,,We can just take the square\Nroot of both sides of this.
Dialogue: 0,0:05:57.39,0:06:00.55,Default,,0000,0000,0000,,And if we take the square root\Nof both sides what do we get?
Dialogue: 0,0:06:00.55,0:06:03.61,Default,,0000,0000,0000,,We get-- once again,\Narbitrarily switching colors--
Dialogue: 0,0:06:03.61,0:06:09.23,Default,,0000,0000,0000,,that x plus 8 is equal to, and\Nremember this, the plus or
Dialogue: 0,0:06:09.23,0:06:12.88,Default,,0000,0000,0000,,minus square root of 121.
Dialogue: 0,0:06:12.88,0:06:14.59,Default,,0000,0000,0000,,And what's the\Nsquare root of 121?
Dialogue: 0,0:06:14.59,0:06:15.96,Default,,0000,0000,0000,,Well it's 11, right?
Dialogue: 0,0:06:15.96,0:06:17.63,Default,,0000,0000,0000,,So then we come here.
Dialogue: 0,0:06:17.63,0:06:18.80,Default,,0000,0000,0000,,Let me box this away.
Dialogue: 0,0:06:18.80,0:06:20.62,Default,,0000,0000,0000,,This was just an aside.
Dialogue: 0,0:06:20.62,0:06:26.83,Default,,0000,0000,0000,,So we get x plus 8 is equal\Nto plus or minus 11.
Dialogue: 0,0:06:26.83,0:06:30.42,Default,,0000,0000,0000,,And so x is equal to-- subtract\N8 from both sides-- minus
Dialogue: 0,0:06:30.42,0:06:33.86,Default,,0000,0000,0000,,8 plus or minus 11.
Dialogue: 0,0:06:33.86,0:06:41.59,Default,,0000,0000,0000,,And so x could equal-- so\Nminus 8 plus 11 is 3.
Dialogue: 0,0:06:41.59,0:06:41.97,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:06:44.80,0:06:48.16,Default,,0000,0000,0000,,Let me make sure I\Ndid that right.
Dialogue: 0,0:06:48.16,0:06:53.31,Default,,0000,0000,0000,,x is equal to minus\N8 plus or minus 11.
Dialogue: 0,0:06:53.31,0:06:54.14,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:06:54.14,0:06:55.35,Default,,0000,0000,0000,,That's right.
Dialogue: 0,0:06:55.35,0:06:59.27,Default,,0000,0000,0000,,So x could be equal to 3.
Dialogue: 0,0:06:59.27,0:07:02.96,Default,,0000,0000,0000,,And then if I took minus\N8 minus 11, x could
Dialogue: 0,0:07:02.96,0:07:10.42,Default,,0000,0000,0000,,also equal minus 19.
Dialogue: 0,0:07:10.42,0:07:11.35,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:07:11.35,0:07:13.20,Default,,0000,0000,0000,,And let's see if\Nthat makes sense.
Dialogue: 0,0:07:13.20,0:07:18.68,Default,,0000,0000,0000,,So in theory this should be\Nable to be factored as x
Dialogue: 0,0:07:18.68,0:07:23.77,Default,,0000,0000,0000,,minus 3 times x plus\N19 is equal to 0.
Dialogue: 0,0:07:23.77,0:07:24.03,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:07:24.03,0:07:26.16,Default,,0000,0000,0000,,Because these are the two\Nsolutions of this equation.
Dialogue: 0,0:07:26.16,0:07:28.19,Default,,0000,0000,0000,,And that works out, right?
Dialogue: 0,0:07:28.19,0:07:31.34,Default,,0000,0000,0000,,Minus 3 times 19 is minus 57.
Dialogue: 0,0:07:31.34,0:07:36.92,Default,,0000,0000,0000,,And minus 3 plus\N19 is plus 16x.
Dialogue: 0,0:07:36.92,0:07:39.12,Default,,0000,0000,0000,,We could have just immediately\Nfactored it this way, but if
Dialogue: 0,0:07:39.12,0:07:41.03,Default,,0000,0000,0000,,that wasn't obvious to us--\Nbecause, you know, at least
Dialogue: 0,0:07:41.03,0:07:43.60,Default,,0000,0000,0000,,19 is kind of a strange\Nnumber-- we could do it
Dialogue: 0,0:07:43.60,0:07:46.80,Default,,0000,0000,0000,,completing the square.
Dialogue: 0,0:07:46.80,0:07:47.69,Default,,0000,0000,0000,,And so why is it called\Ncompleting the square?
Dialogue: 0,0:07:47.69,0:07:49.92,Default,,0000,0000,0000,,Because you get it in this form\Nand then you have to add this
Dialogue: 0,0:07:49.92,0:07:52.95,Default,,0000,0000,0000,,64 here to kind of complete the\Nsquare-- to turn this
Dialogue: 0,0:07:52.95,0:07:56.02,Default,,0000,0000,0000,,left-hand expression into\Na squared expression.
Dialogue: 0,0:07:56.02,0:07:56.77,Default,,0000,0000,0000,,Let's do one more.
Dialogue: 0,0:07:56.77,0:07:59.92,Default,,0000,0000,0000,,And I'll do less explanation\Nand more just chugging through
Dialogue: 0,0:07:59.92,0:08:02.10,Default,,0000,0000,0000,,the problem, and that actually\Nmight make it seem simpler.
Dialogue: 0,0:08:04.80,0:08:07.08,Default,,0000,0000,0000,,But this is going to\Nbe a hairier problem.
Dialogue: 0,0:08:07.08,0:08:19.93,Default,,0000,0000,0000,,So let's say I have 6x squared\Nminus 7x minus 3 is equal to 0.
Dialogue: 0,0:08:19.93,0:08:22.98,Default,,0000,0000,0000,,You could try to factor it,\Nbut personally I don't
Dialogue: 0,0:08:22.98,0:08:25.26,Default,,0000,0000,0000,,enjoy factoring things\Nwhen I have a coefficient.
Dialogue: 0,0:08:25.26,0:08:27.59,Default,,0000,0000,0000,,And you can say, oh well why\Ndon't we divide both sides
Dialogue: 0,0:08:27.59,0:08:28.97,Default,,0000,0000,0000,,of this equation by 6?
Dialogue: 0,0:08:28.97,0:08:30.96,Default,,0000,0000,0000,,But then you'd get a fraction\Nhere and a fraction here.
Dialogue: 0,0:08:30.96,0:08:33.58,Default,,0000,0000,0000,,And that's even worse to\Nfactor just by inspection.
Dialogue: 0,0:08:33.58,0:08:35.19,Default,,0000,0000,0000,,You could do the\Nquadratic equation.
Dialogue: 0,0:08:35.19,0:08:37.31,Default,,0000,0000,0000,,And maybe I'll show you in a\Nfuture video, the quadratic
Dialogue: 0,0:08:37.31,0:08:39.50,Default,,0000,0000,0000,,equation-- and I think I've\Nalready done one where I proved
Dialogue: 0,0:08:39.50,0:08:40.63,Default,,0000,0000,0000,,the quadratic equation.
Dialogue: 0,0:08:40.63,0:08:42.38,Default,,0000,0000,0000,,But the quadratic\Nequation is essentially
Dialogue: 0,0:08:42.38,0:08:43.17,Default,,0000,0000,0000,,completing the square.
Dialogue: 0,0:08:43.17,0:08:44.09,Default,,0000,0000,0000,,It's kind of a shortcut.
Dialogue: 0,0:08:44.09,0:08:46.28,Default,,0000,0000,0000,,It's just kind of\Nremembering the formula.
Dialogue: 0,0:08:46.28,0:08:48.32,Default,,0000,0000,0000,,But let's complete the square\Nhere, because that's what the
Dialogue: 0,0:08:48.32,0:08:50.64,Default,,0000,0000,0000,,point of this video was.
Dialogue: 0,0:08:50.64,0:08:54.65,Default,,0000,0000,0000,,So let's add the 3 to both\Nsides of that equation.
Dialogue: 0,0:08:54.65,0:08:56.30,Default,,0000,0000,0000,,We could do-- well,\Nlet's add the 3 first.
Dialogue: 0,0:08:56.30,0:09:04.82,Default,,0000,0000,0000,,So you get 6 x squared\Nminus 7x is equal to 3.
Dialogue: 0,0:09:04.82,0:09:06.77,Default,,0000,0000,0000,,I added 3 to both sides.
Dialogue: 0,0:09:06.77,0:09:09.47,Default,,0000,0000,0000,,And some teachers will leave\Nthe minus 3 here, and then try
Dialogue: 0,0:09:09.47,0:09:11.05,Default,,0000,0000,0000,,to figure out what to add\Nto it and all of that.
Dialogue: 0,0:09:11.05,0:09:13.17,Default,,0000,0000,0000,,But I like to get it out of the\Nway so that I can figure out
Dialogue: 0,0:09:13.17,0:09:16.08,Default,,0000,0000,0000,,very clearly what number\NI should put here.
Dialogue: 0,0:09:16.08,0:09:18.23,Default,,0000,0000,0000,,But I also don't\Nlike the 6 here.
Dialogue: 0,0:09:18.23,0:09:19.55,Default,,0000,0000,0000,,It just complicates things.
Dialogue: 0,0:09:19.55,0:09:25.99,Default,,0000,0000,0000,,I like to have it x plus a\Nsquared, not some square root
Dialogue: 0,0:09:25.99,0:09:27.45,Default,,0000,0000,0000,,coefficient on the x term.
Dialogue: 0,0:09:27.45,0:09:31.53,Default,,0000,0000,0000,,So let's divide both sides of\Nthis equation by 6, and you get
Dialogue: 0,0:09:31.53,0:09:39.73,Default,,0000,0000,0000,,x squared minus 7/6 x is equal\Nto-- 3 divided by 6
Dialogue: 0,0:09:39.73,0:09:41.57,Default,,0000,0000,0000,,is equal to 1/2.
Dialogue: 0,0:09:41.57,0:09:43.19,Default,,0000,0000,0000,,And we could have made\Nthat our first step.
Dialogue: 0,0:09:43.19,0:09:46.45,Default,,0000,0000,0000,,We could have divided by 6\Nright at that first step.
Dialogue: 0,0:09:46.45,0:09:49.25,Default,,0000,0000,0000,,Anyway, now let's try to\Ncomplete the square.
Dialogue: 0,0:09:49.25,0:09:51.80,Default,,0000,0000,0000,,So we have x squared-- I'm just\Ngoing to open up some space--
Dialogue: 0,0:09:51.80,0:09:59.53,Default,,0000,0000,0000,,minus 7/6 x plus something is\Ngoing to be equal to 1/2.
Dialogue: 0,0:09:59.53,0:10:02.40,Default,,0000,0000,0000,,And so we have to add something\Nhere so that this left-hand
Dialogue: 0,0:10:02.40,0:10:05.29,Default,,0000,0000,0000,,expression becomes a\Nsquared expression.
Dialogue: 0,0:10:05.29,0:10:06.62,Default,,0000,0000,0000,,So how do we do that?
Dialogue: 0,0:10:06.62,0:10:10.77,Default,,0000,0000,0000,,Well essentially we look at\Nthis coefficient, and keep
Dialogue: 0,0:10:10.77,0:10:14.61,Default,,0000,0000,0000,,in mind this is not just\N7/6 it's minus 7/6.
Dialogue: 0,0:10:14.61,0:10:17.46,Default,,0000,0000,0000,,You take 1/2 of it, and\Nthen you square it.
Dialogue: 0,0:10:17.46,0:10:18.61,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:10:18.61,0:10:19.69,Default,,0000,0000,0000,,Let me do it.
Dialogue: 0,0:10:19.69,0:10:25.29,Default,,0000,0000,0000,,x plus a, squared, is\Nequal to x squared plus
Dialogue: 0,0:10:25.29,0:10:28.82,Default,,0000,0000,0000,,2ax plus a squared.
Dialogue: 0,0:10:28.82,0:10:29.07,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:10:29.07,0:10:30.75,Default,,0000,0000,0000,,This is what you have to\Nremember all the time.
Dialogue: 0,0:10:30.75,0:10:33.56,Default,,0000,0000,0000,,That's all completing the\Nsquare is based off of.
Dialogue: 0,0:10:33.56,0:10:34.98,Default,,0000,0000,0000,,So what did I say just now?
Dialogue: 0,0:10:34.98,0:10:37.26,Default,,0000,0000,0000,,Well, this term is going\Nto be 1/2 of this
Dialogue: 0,0:10:37.26,0:10:39.19,Default,,0000,0000,0000,,coefficient squared.
Dialogue: 0,0:10:39.19,0:10:40.19,Default,,0000,0000,0000,,And how do we know that?
Dialogue: 0,0:10:40.19,0:10:43.88,Default,,0000,0000,0000,,Because a is going to be 1/2 of\Nthis coefficient if you just
Dialogue: 0,0:10:43.88,0:10:45.85,Default,,0000,0000,0000,,do a little bit of\Npattern matching.
Dialogue: 0,0:10:45.85,0:10:48.76,Default,,0000,0000,0000,,So what's 1/2 of\Nthis coefficient?
Dialogue: 0,0:10:48.76,0:10:54.05,Default,,0000,0000,0000,,1/2 of minus 7/6 is minus 7/12.
Dialogue: 0,0:10:54.05,0:10:56.64,Default,,0000,0000,0000,,So if you want you could\Nwrite a equals minus
Dialogue: 0,0:10:56.64,0:10:58.77,Default,,0000,0000,0000,,7/12 for our example.
Dialogue: 0,0:10:58.77,0:11:00.77,Default,,0000,0000,0000,,And I just multiplied\Nthis by 1/2.
Dialogue: 0,0:11:00.77,0:11:01.98,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:11:01.98,0:11:03.66,Default,,0000,0000,0000,,So what do I add to both sides?
Dialogue: 0,0:11:03.66,0:11:06.03,Default,,0000,0000,0000,,I add a squared.
Dialogue: 0,0:11:06.03,0:11:08.93,Default,,0000,0000,0000,,So what's 7/12 squared?
Dialogue: 0,0:11:08.93,0:11:13.22,Default,,0000,0000,0000,,Well that's going to be 49/144.
Dialogue: 0,0:11:13.22,0:11:15.00,Default,,0000,0000,0000,,If I did it to the left-hand\Nside I have to do it to
Dialogue: 0,0:11:15.00,0:11:16.63,Default,,0000,0000,0000,,the right-hand side.
Dialogue: 0,0:11:16.63,0:11:22.24,Default,,0000,0000,0000,,Plus 49/144.
Dialogue: 0,0:11:22.24,0:11:26.12,Default,,0000,0000,0000,,And now how can I simplify\Nthis left-hand side?
Dialogue: 0,0:11:26.12,0:11:26.88,Default,,0000,0000,0000,,What's our next step?
Dialogue: 0,0:11:26.88,0:11:28.47,Default,,0000,0000,0000,,Well we now know it\Nis a perfect square.
Dialogue: 0,0:11:28.47,0:11:31.55,Default,,0000,0000,0000,,In fact, we know what a\Nis. a is minus 7/12.
Dialogue: 0,0:11:31.55,0:11:35.20,Default,,0000,0000,0000,,And so we know that this\Nleft-hand side of this equation
Dialogue: 0,0:11:35.20,0:11:43.39,Default,,0000,0000,0000,,is x minus a-- or x plus a,\Nbut a is a negative number.
Dialogue: 0,0:11:43.39,0:11:47.98,Default,,0000,0000,0000,,So x plus a, and a is\Nnegative, squared.
Dialogue: 0,0:11:47.98,0:11:50.35,Default,,0000,0000,0000,,And if you want you can\Nmultiply this out and confirm
Dialogue: 0,0:11:50.35,0:11:53.13,Default,,0000,0000,0000,,that it truly equals this.
Dialogue: 0,0:11:53.13,0:11:55.92,Default,,0000,0000,0000,,And that is going to be equal\Nto-- let's get a common
Dialogue: 0,0:11:55.92,0:11:58.36,Default,,0000,0000,0000,,denominator, 144.
Dialogue: 0,0:11:58.36,0:12:04.07,Default,,0000,0000,0000,,So 72 plus 49 equals 121.
Dialogue: 0,0:12:04.07,0:12:06.30,Default,,0000,0000,0000,,121/144.
Dialogue: 0,0:12:06.30,0:12:09.21,Default,,0000,0000,0000,,So we have x minus 7/12,\Nall of that squared
Dialogue: 0,0:12:09.21,0:12:13.18,Default,,0000,0000,0000,,is equal to 121/144.
Dialogue: 0,0:12:13.18,0:12:14.30,Default,,0000,0000,0000,,So what do we do now?
Dialogue: 0,0:12:14.30,0:12:15.57,Default,,0000,0000,0000,,Well now we just take\Nthe square root of both
Dialogue: 0,0:12:15.57,0:12:17.70,Default,,0000,0000,0000,,sides of this equation.
Dialogue: 0,0:12:17.70,0:12:20.14,Default,,0000,0000,0000,,And I'm trying to\Nfree up some space.
Dialogue: 0,0:12:20.14,0:12:22.22,Default,,0000,0000,0000,,Switch to green.
Dialogue: 0,0:12:22.22,0:12:25.32,Default,,0000,0000,0000,,Let me partition this off.
Dialogue: 0,0:12:25.32,0:12:33.31,Default,,0000,0000,0000,,And we get x minus 7/12 is\Nequal to the plus or minus
Dialogue: 0,0:12:33.31,0:12:33.94,Default,,0000,0000,0000,,square root of that.
Dialogue: 0,0:12:33.94,0:12:38.12,Default,,0000,0000,0000,,So plus or minus 11/12.
Dialogue: 0,0:12:38.12,0:12:38.39,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:12:38.39,0:12:39.66,Default,,0000,0000,0000,,Square root of 121 is 11.
Dialogue: 0,0:12:39.66,0:12:42.42,Default,,0000,0000,0000,,Square root of 144 is 12.
Dialogue: 0,0:12:42.42,0:12:44.48,Default,,0000,0000,0000,,So then we could add 7/12 to\Nboth sides of this equation,
Dialogue: 0,0:12:44.48,0:12:53.10,Default,,0000,0000,0000,,and we get x is equal to\N7/12 plus or minus 11/12.
Dialogue: 0,0:12:53.10,0:12:58.66,Default,,0000,0000,0000,,Well that equals 7\Nplus or minus 11/12.
Dialogue: 0,0:12:58.66,0:13:00.05,Default,,0000,0000,0000,,So what are the two options?
Dialogue: 0,0:13:00.05,0:13:03.93,Default,,0000,0000,0000,,7 plus 11 is 18, over 12.
Dialogue: 0,0:13:03.93,0:13:08.21,Default,,0000,0000,0000,,So x could equal 18/12, is 3/2.
Dialogue: 0,0:13:08.21,0:13:11.01,Default,,0000,0000,0000,,Or, what's 7 minus 11?
Dialogue: 0,0:13:11.01,0:13:12.76,Default,,0000,0000,0000,,That's minus 4/12.
Dialogue: 0,0:13:12.76,0:13:15.37,Default,,0000,0000,0000,,So it's minus 1/3.
Dialogue: 0,0:13:15.37,0:13:16.63,Default,,0000,0000,0000,,There you have it.
Dialogue: 0,0:13:16.63,0:13:17.94,Default,,0000,0000,0000,,That is completing the square.
Dialogue: 0,0:13:17.94,0:13:20.22,Default,,0000,0000,0000,,Hopefully you found that\Nreasonably insightful.
Dialogue: 0,0:13:20.22,0:13:23.34,Default,,0000,0000,0000,,And if you want to prove the\Nquadratic equation, all you
Dialogue: 0,0:13:23.34,0:13:27.32,Default,,0000,0000,0000,,have to do is instead of having\Nnumbers here, write a x squared
Dialogue: 0,0:13:27.32,0:13:29.82,Default,,0000,0000,0000,,plus bx plus c equals 0.
Dialogue: 0,0:13:29.82,0:13:34.13,Default,,0000,0000,0000,,And then complete the square\Nusing the a, b, and c's
Dialogue: 0,0:13:34.13,0:13:35.06,Default,,0000,0000,0000,,instead of numbers.
Dialogue: 0,0:13:35.06,0:13:37.18,Default,,0000,0000,0000,,And you will end up with\Nthe quadratic equation
Dialogue: 0,0:13:37.18,0:13:38.11,Default,,0000,0000,0000,,by this point.
Dialogue: 0,0:13:38.11,0:13:39.51,Default,,0000,0000,0000,,And I think I did\Nthat in a video.
Dialogue: 0,0:13:39.51,0:13:41.60,Default,,0000,0000,0000,,Let me know if I didn't\Nand I'll do it for you.
Dialogue: 0,0:13:41.60,0:13:44.54,Default,,0000,0000,0000,,Anyway, I'll see you\Nin the next video.