[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.87,0:00:07.22,Default,,0000,0000,0000,,Going to have a look at a very\Nsimple process. It's called
Dialogue: 0,0:00:07.22,0:00:08.56,Default,,0000,0000,0000,,completing the square.
Dialogue: 0,0:00:09.22,0:00:14.41,Default,,0000,0000,0000,,In order to get to it and to\Nshow its potential use, I want
Dialogue: 0,0:00:14.41,0:00:18.12,Default,,0000,0000,0000,,to start with a simple equation.\NX squared equals 9.
Dialogue: 0,0:00:19.90,0:00:24.66,Default,,0000,0000,0000,,In order to find out what taxes\Nwe would take the square root of
Dialogue: 0,0:00:24.66,0:00:29.42,Default,,0000,0000,0000,,both sides, so the square root\Nof X squared is just X and the
Dialogue: 0,0:00:29.42,0:00:32.82,Default,,0000,0000,0000,,square root of 9 is plus three\Nor minus three.
Dialogue: 0,0:00:33.35,0:00:38.36,Default,,0000,0000,0000,,Be'cause minus three all squared\Nis also 9.
Dialogue: 0,0:00:38.87,0:00:43.59,Default,,0000,0000,0000,,So that was relatively\Nstraightforward. Both of these
Dialogue: 0,0:00:43.59,0:00:46.54,Default,,0000,0000,0000,,two numbers were square numbers,
Dialogue: 0,0:00:46.54,0:00:49.56,Default,,0000,0000,0000,,complete squares. What if we
Dialogue: 0,0:00:49.56,0:00:53.29,Default,,0000,0000,0000,,got? X squared is equal to
Dialogue: 0,0:00:53.29,0:00:59.75,Default,,0000,0000,0000,,5. Do the same again, so\Nwe take the square root of
Dialogue: 0,0:00:59.75,0:01:05.28,Default,,0000,0000,0000,,each side X equals. Now 5\Nis not a square number, it
Dialogue: 0,0:01:05.28,0:01:08.97,Default,,0000,0000,0000,,does not have an exact\Nsquare root, but
Dialogue: 0,0:01:08.97,0:01:14.04,Default,,0000,0000,0000,,nevertheless we can write\Nit as root 5 that is exact
Dialogue: 0,0:01:14.04,0:01:15.89,Default,,0000,0000,0000,,or minus Route 5.
Dialogue: 0,0:01:17.10,0:01:21.94,Default,,0000,0000,0000,,So far so good. The same process\Nis working each time.
Dialogue: 0,0:01:22.71,0:01:27.54,Default,,0000,0000,0000,,Let's have a look at something\Nnow, like X minus 7.
Dialogue: 0,0:01:28.19,0:01:31.09,Default,,0000,0000,0000,,All squared equals 3.
Dialogue: 0,0:01:31.94,0:01:36.86,Default,,0000,0000,0000,,How can we solve this? Well\Nagain, this side of the
Dialogue: 0,0:01:36.86,0:01:41.33,Default,,0000,0000,0000,,equation. We've got something\Ncalled a complete square, so we
Dialogue: 0,0:01:41.33,0:01:47.58,Default,,0000,0000,0000,,can take the square root of each\Nside. X minus Seven is equal to
Dialogue: 0,0:01:47.58,0:01:54.29,Default,,0000,0000,0000,,the square root of 3 or minus\Nthe square root of 3, and so we
Dialogue: 0,0:01:54.29,0:02:01.44,Default,,0000,0000,0000,,can now add 7 to each of these.\NSo X is equal to 7 Plus Route
Dialogue: 0,0:02:01.44,0:02:02.78,Default,,0000,0000,0000,,3 or 7.
Dialogue: 0,0:02:02.83,0:02:04.61,Default,,0000,0000,0000,,Minus Route 3.
Dialogue: 0,0:02:06.88,0:02:14.61,Default,,0000,0000,0000,,I just have a look at another\NOne X plus three, all squared is
Dialogue: 0,0:02:14.61,0:02:16.26,Default,,0000,0000,0000,,equal to 5.
Dialogue: 0,0:02:18.67,0:02:24.75,Default,,0000,0000,0000,,Again, this is a complete\Nsquare, so again we can take the
Dialogue: 0,0:02:24.75,0:02:31.85,Default,,0000,0000,0000,,square root X +3 is equal to\NRoute 5 or minus Route 5. Now
Dialogue: 0,0:02:31.85,0:02:38.44,Default,,0000,0000,0000,,to get X on its own, we\Nneed to take three away from
Dialogue: 0,0:02:38.44,0:02:45.54,Default,,0000,0000,0000,,each side, so we have X equals\Nminus 3 + 5 or minus 3
Dialogue: 0,0:02:45.54,0:02:47.06,Default,,0000,0000,0000,,minus Route 5.
Dialogue: 0,0:02:51.01,0:02:57.74,Default,,0000,0000,0000,,What if we got\NX squared plus six
Dialogue: 0,0:02:57.74,0:03:00.26,Default,,0000,0000,0000,,X equals 4?
Dialogue: 0,0:03:01.59,0:03:07.44,Default,,0000,0000,0000,,Problem is this X squared plus\N6X is not a complete square, so
Dialogue: 0,0:03:07.44,0:03:10.14,Default,,0000,0000,0000,,we can't just take the square
Dialogue: 0,0:03:10.14,0:03:15.56,Default,,0000,0000,0000,,root. So in terms of handling\Nsomething like this, we've got
Dialogue: 0,0:03:15.56,0:03:18.46,Default,,0000,0000,0000,,to have a way of getting a
Dialogue: 0,0:03:18.46,0:03:22.08,Default,,0000,0000,0000,,complete square. So the\Nprocess that we're going to
Dialogue: 0,0:03:22.08,0:03:24.42,Default,,0000,0000,0000,,be looking at it's called\Ncompleting the square.
Dialogue: 0,0:03:25.57,0:03:31.33,Default,,0000,0000,0000,,The sorts of expressions that we\Nhave before will like this X
Dialogue: 0,0:03:31.33,0:03:33.25,Default,,0000,0000,0000,,plus a all squared.
Dialogue: 0,0:03:34.03,0:03:40.40,Default,,0000,0000,0000,,Or X minus a all squared, so\Nthat's what a complete square
Dialogue: 0,0:03:40.40,0:03:43.59,Default,,0000,0000,0000,,looks like. One of these two.
Dialogue: 0,0:03:44.15,0:03:51.04,Default,,0000,0000,0000,,So let's multiply this out and\Nsee what we get. So this is X
Dialogue: 0,0:03:51.04,0:03:54.48,Default,,0000,0000,0000,,plus a times by X plus A.
Dialogue: 0,0:03:55.96,0:04:00.16,Default,,0000,0000,0000,,And we do X times by X. That\Ngives us X squared.
Dialogue: 0,0:04:00.93,0:04:05.43,Default,,0000,0000,0000,,And we do a Times by X, which\Ngives us a X.
Dialogue: 0,0:04:06.15,0:04:12.10,Default,,0000,0000,0000,,And then with X times by a,\Nwhich again gives us a X and
Dialogue: 0,0:04:12.10,0:04:18.48,Default,,0000,0000,0000,,then at the end a Times by a,\Nwhich gives us a squared. And so
Dialogue: 0,0:04:18.48,0:04:21.45,Default,,0000,0000,0000,,we've X squared plus 2X plus A
Dialogue: 0,0:04:21.45,0:04:28.100,Default,,0000,0000,0000,,squared. I can do the same\Nwith this One X minus a Times by
Dialogue: 0,0:04:28.100,0:04:35.66,Default,,0000,0000,0000,,X minus A and it's going to give\Nvery similar results X times by
Dialogue: 0,0:04:35.66,0:04:42.80,Default,,0000,0000,0000,,X will give me X squared X times\Nby minus A minus 8X minus 8
Dialogue: 0,0:04:42.80,0:04:49.47,Default,,0000,0000,0000,,times by X minus 8X minus 8\Ntimes by minus A plus a squared.
Dialogue: 0,0:04:49.47,0:04:52.32,Default,,0000,0000,0000,,So tidying up the two middle
Dialogue: 0,0:04:52.32,0:04:58.43,Default,,0000,0000,0000,,bits. Minus X minus X, minus\N2X and then plus A squared.
Dialogue: 0,0:04:58.43,0:05:03.53,Default,,0000,0000,0000,,So this is what complete\Nsquares look like. They look
Dialogue: 0,0:05:03.53,0:05:06.08,Default,,0000,0000,0000,,like one of these two.
Dialogue: 0,0:05:07.21,0:05:13.57,Default,,0000,0000,0000,,Well. Can I make this\Nlook like a complete square in
Dialogue: 0,0:05:13.57,0:05:19.39,Default,,0000,0000,0000,,some way shape or form? If I\Ncompare this with this, what is
Dialogue: 0,0:05:19.39,0:05:21.19,Default,,0000,0000,0000,,it that I see?
Dialogue: 0,0:05:21.85,0:05:27.24,Default,,0000,0000,0000,,Well, perhaps one of things I\Nmight like to have is this
Dialogue: 0,0:05:27.24,0:05:32.63,Default,,0000,0000,0000,,written as just a function X\Nsquared plus 6X minus four? And
Dialogue: 0,0:05:32.63,0:05:37.56,Default,,0000,0000,0000,,let's not worry too much about\Nsolving an equation. What we
Dialogue: 0,0:05:37.56,0:05:42.06,Default,,0000,0000,0000,,want to concentrate on is this\Nprocess of completing the
Dialogue: 0,0:05:42.06,0:05:46.54,Default,,0000,0000,0000,,square, so I'm going to take\Nthis quadratic function X
Dialogue: 0,0:05:46.54,0:05:52.38,Default,,0000,0000,0000,,squared plus 6X minus four, and\NI'm going to compare it with the
Dialogue: 0,0:05:52.38,0:05:58.53,Default,,0000,0000,0000,,complete square. X squared\Nplus 2X plus
Dialogue: 0,0:05:58.53,0:06:04.01,Default,,0000,0000,0000,,A squared. Now The\NX squared so the same.
Dialogue: 0,0:06:05.21,0:06:11.76,Default,,0000,0000,0000,,6X2A X I've got to have these\Ntwo terms the same. They've got
Dialogue: 0,0:06:11.76,0:06:17.81,Default,,0000,0000,0000,,to match. They've got to be\Nexactly the same, and that means
Dialogue: 0,0:06:17.81,0:06:21.34,Default,,0000,0000,0000,,that the six has to be equal
Dialogue: 0,0:06:21.34,0:06:28.79,Default,,0000,0000,0000,,to 2A. And of course,\Nthat tells us that the A
Dialogue: 0,0:06:28.79,0:06:31.38,Default,,0000,0000,0000,,is equal to 3.
Dialogue: 0,0:06:32.75,0:06:39.43,Default,,0000,0000,0000,,So if I make a equal to\Nthree, then I've got plus A
Dialogue: 0,0:06:39.43,0:06:42.00,Default,,0000,0000,0000,,squared on the end +9.
Dialogue: 0,0:06:43.30,0:06:49.36,Default,,0000,0000,0000,,So. I can look at this first bit\Nand I can make it equal to that.
Dialogue: 0,0:06:50.07,0:06:56.80,Default,,0000,0000,0000,,So let's write this down\NX squared plus 6X minus
Dialogue: 0,0:06:56.80,0:07:03.93,Default,,0000,0000,0000,,4 equals. X plus\Nthree all squared. Remember this
Dialogue: 0,0:07:03.93,0:07:10.85,Default,,0000,0000,0000,,is X plus three all\Nsquared. Now I'm replacing the
Dialogue: 0,0:07:10.85,0:07:12.93,Default,,0000,0000,0000,,A by three.
Dialogue: 0,0:07:13.83,0:07:18.95,Default,,0000,0000,0000,,Now what more of I got? Well,\NI've added on a squared, so I've
Dialogue: 0,0:07:18.95,0:07:24.44,Default,,0000,0000,0000,,added on 9. So I've got some how\Nto get rid of that. Well, let's
Dialogue: 0,0:07:24.44,0:07:26.64,Default,,0000,0000,0000,,just take it away, minus 3
Dialogue: 0,0:07:26.64,0:07:31.82,Default,,0000,0000,0000,,squared. And then I can keep\Nthis minus four at the end as it
Dialogue: 0,0:07:31.82,0:07:39.57,Default,,0000,0000,0000,,was. So now I've X plus\Nthree all squared minus 9 -
Dialogue: 0,0:07:39.57,0:07:45.85,Default,,0000,0000,0000,,4 gives me X plus three\Nall squared minus 30.
Dialogue: 0,0:07:47.45,0:07:52.44,Default,,0000,0000,0000,,So I have completed the square.\NI made this bit.
Dialogue: 0,0:07:53.79,0:07:57.57,Default,,0000,0000,0000,,Part of a complete square.
Dialogue: 0,0:07:58.45,0:08:03.50,Default,,0000,0000,0000,,And I've done it by comparing\Nthe coefficient of X.
Dialogue: 0,0:08:04.04,0:08:10.27,Default,,0000,0000,0000,,With one of the two standard\Nforms and I saw that what I had
Dialogue: 0,0:08:10.27,0:08:14.28,Default,,0000,0000,0000,,to do was take half the\Ncoefficient of X.
Dialogue: 0,0:08:15.75,0:08:22.90,Default,,0000,0000,0000,,Let's have a look then at\Nanother example, X squared minus
Dialogue: 0,0:08:22.90,0:08:30.70,Default,,0000,0000,0000,,8X plus Seven and I want\Nto write this so it's got
Dialogue: 0,0:08:30.70,0:08:33.95,Default,,0000,0000,0000,,a complete square in it.
Dialogue: 0,0:08:34.64,0:08:39.77,Default,,0000,0000,0000,,Well, one of the standard\Nforms for the complete square
Dialogue: 0,0:08:39.77,0:08:45.41,Default,,0000,0000,0000,,that we had was X squared\Nminus 2X plus A squared.
Dialogue: 0,0:08:47.90,0:08:50.99,Default,,0000,0000,0000,,And I want to make these two
Dialogue: 0,0:08:50.99,0:08:52.83,Default,,0000,0000,0000,,terms. The same.
Dialogue: 0,0:08:54.10,0:08:57.95,Default,,0000,0000,0000,,So again, we can see that the A.
Dialogue: 0,0:08:58.80,0:09:05.46,Default,,0000,0000,0000,,Has got to be 4 because minus 8\Nis minus 2 times by 4.
Dialogue: 0,0:09:08.09,0:09:15.52,Default,,0000,0000,0000,,So we've got X squared\Nminus 8X plus Seven is
Dialogue: 0,0:09:15.52,0:09:21.93,Default,,0000,0000,0000,,equal to. X minus\Nfour all squared.
Dialogue: 0,0:09:22.89,0:09:27.95,Default,,0000,0000,0000,,So I've ensured I've got the X\Nsquared. I've ensured that I've
Dialogue: 0,0:09:27.95,0:09:33.86,Default,,0000,0000,0000,,got the minus 8X, but I've also\Nadded on a squared, so I've got
Dialogue: 0,0:09:33.86,0:09:39.35,Default,,0000,0000,0000,,a squared too much, so I must\Ntake away 4 squared and then
Dialogue: 0,0:09:39.35,0:09:45.68,Default,,0000,0000,0000,,I've got the 7:00 that I need to\Nadd on to keep the equal sign.
Dialogue: 0,0:09:46.31,0:09:52.54,Default,,0000,0000,0000,,And so this is now\NX minus four all
Dialogue: 0,0:09:52.54,0:09:58.77,Default,,0000,0000,0000,,squared minus 16 +\N7 X minus four all
Dialogue: 0,0:09:58.77,0:10:00.84,Default,,0000,0000,0000,,squared minus 9.
Dialogue: 0,0:10:02.14,0:10:06.09,Default,,0000,0000,0000,,Let's take one more example.
Dialogue: 0,0:10:06.11,0:10:08.35,Default,,0000,0000,0000,,X
Dialogue: 0,0:10:08.35,0:10:14.98,Default,,0000,0000,0000,,squared Plus 5X\Nplus three and let's see if we
Dialogue: 0,0:10:14.98,0:10:19.61,Default,,0000,0000,0000,,can follow this one through\Nwithout having to write down the
Dialogue: 0,0:10:19.61,0:10:23.40,Default,,0000,0000,0000,,comparison. In other words, by\Ndoing it by inspection.
Dialogue: 0,0:10:24.05,0:10:30.34,Default,,0000,0000,0000,,What do we need? We need a\Ncomplete square, so we need X
Dialogue: 0,0:10:30.34,0:10:37.12,Default,,0000,0000,0000,,and we look at this number here.\NThe coefficient of the X turn on
Dialogue: 0,0:10:37.12,0:10:39.05,Default,,0000,0000,0000,,this left hand side.
Dialogue: 0,0:10:39.77,0:10:45.15,Default,,0000,0000,0000,,We want half that coefficient,\Nso we want five over 2 and we've
Dialogue: 0,0:10:45.15,0:10:50.53,Default,,0000,0000,0000,,got a plus sign, so it's got to\Nbe X +5 over 2.
Dialogue: 0,0:10:51.22,0:10:57.38,Default,,0000,0000,0000,,If we were to multiply out this\Nbracket, we would be adding on
Dialogue: 0,0:10:57.38,0:11:03.54,Default,,0000,0000,0000,,an additional A squared where\Nfive over 2 is the a. So we've
Dialogue: 0,0:11:03.54,0:11:10.18,Default,,0000,0000,0000,,got to take that away. Takeaway\N5 over 2 squared and then at the
Dialogue: 0,0:11:10.18,0:11:12.55,Default,,0000,0000,0000,,end we've got plus 3.
Dialogue: 0,0:11:13.34,0:11:19.81,Default,,0000,0000,0000,,So this gives us\NX +5 over 2
Dialogue: 0,0:11:19.81,0:11:26.28,Default,,0000,0000,0000,,or squared minus 25\Nover 4 + 3.
Dialogue: 0,0:11:27.33,0:11:33.52,Default,,0000,0000,0000,,And of course, we'd like to\Ncombine these numbers at this
Dialogue: 0,0:11:33.52,0:11:41.40,Default,,0000,0000,0000,,end X +5 over 2 all squared\Nminus 25 over 4 plus. Now we
Dialogue: 0,0:11:41.40,0:11:48.16,Default,,0000,0000,0000,,need to convert these to\Nquarters as well, so three is 12
Dialogue: 0,0:11:48.16,0:11:54.74,Default,,0000,0000,0000,,quarters. Now we can combine\Nthese, since they're both in
Dialogue: 0,0:11:54.74,0:12:02.05,Default,,0000,0000,0000,,terms of quarters X +5 over\N2, all squared minus 13 over
Dialogue: 0,0:12:02.05,0:12:04.48,Default,,0000,0000,0000,,4. And we'd leave you like that.
Dialogue: 0,0:12:05.51,0:12:09.27,Default,,0000,0000,0000,,This process now seems to be\Nworking quite well.
Dialogue: 0,0:12:09.80,0:12:13.14,Default,,0000,0000,0000,,But of course, we haven't\Ndealt with every kind of
Dialogue: 0,0:12:13.14,0:12:16.15,Default,,0000,0000,0000,,quadratic expression we could\Nhave, because so far we've
Dialogue: 0,0:12:16.15,0:12:19.82,Default,,0000,0000,0000,,only had a unit coefficient\Nhere in front of the X
Dialogue: 0,0:12:19.82,0:12:22.83,Default,,0000,0000,0000,,squared. We haven't had\Nanother number like two or
Dialogue: 0,0:12:22.83,0:12:26.83,Default,,0000,0000,0000,,three or whatever, so let's\Nhave a look at what we would
Dialogue: 0,0:12:26.83,0:12:28.17,Default,,0000,0000,0000,,do in that case.
Dialogue: 0,0:12:30.90,0:12:35.21,Default,,0000,0000,0000,,So we have three X\Nsquared.
Dialogue: 0,0:12:36.25,0:12:43.29,Default,,0000,0000,0000,,Minus 9X. Plus\N50, what do we need to do
Dialogue: 0,0:12:43.29,0:12:50.37,Default,,0000,0000,0000,,to begin with? Well, we know how\Nto do this if we've got a
Dialogue: 0,0:12:50.37,0:12:56.44,Default,,0000,0000,0000,,unit coefficient with the X\Nsquared, so let's make it a unit
Dialogue: 0,0:12:56.44,0:13:02.01,Default,,0000,0000,0000,,coefficient by taking out the\Nthree as a common factor. So
Dialogue: 0,0:13:02.01,0:13:08.08,Default,,0000,0000,0000,,that's three brackets X squared,\Nminus 3X. Now 50. What are we
Dialogue: 0,0:13:08.08,0:13:11.12,Default,,0000,0000,0000,,going to do with this? Well?
Dialogue: 0,0:13:11.14,0:13:15.57,Default,,0000,0000,0000,,We divide it by three in order\Nthat when we do the
Dialogue: 0,0:13:15.57,0:13:20.36,Default,,0000,0000,0000,,multiplication 3 * 50 over three\Nwill just give us back the 50.
Dialogue: 0,0:13:20.90,0:13:25.67,Default,,0000,0000,0000,,Now we look at this thing here\Nin the bracket because this is
Dialogue: 0,0:13:25.67,0:13:29.34,Default,,0000,0000,0000,,now exactly the same sort of\Nexpression that we've had
Dialogue: 0,0:13:29.34,0:13:31.82,Default,,0000,0000,0000,,before. Equals 3.
Dialogue: 0,0:13:32.51,0:13:37.02,Default,,0000,0000,0000,,Let's have a big Curly\Nbracket. I'm going to make
Dialogue: 0,0:13:37.02,0:13:41.98,Default,,0000,0000,0000,,this going to complete the\Nsquare around this, so this is
Dialogue: 0,0:13:41.98,0:13:47.84,Default,,0000,0000,0000,,going to be X minus. Look at\Nthe coefficient of X and take
Dialogue: 0,0:13:47.84,0:13:51.45,Default,,0000,0000,0000,,half of it 3 over 2 all\Nsquared.
Dialogue: 0,0:13:52.55,0:13:55.44,Default,,0000,0000,0000,,By doing that, we've added on.
Dialogue: 0,0:13:56.07,0:14:03.71,Default,,0000,0000,0000,,An additional A squared, so we\Nneed to take that off 3
Dialogue: 0,0:14:03.71,0:14:11.36,Default,,0000,0000,0000,,over 2 squared and then finally\Nplus 50 over 3 and close
Dialogue: 0,0:14:11.36,0:14:13.27,Default,,0000,0000,0000,,the big bracket.
Dialogue: 0,0:14:14.82,0:14:22.26,Default,,0000,0000,0000,,3. X minus\Nthree over 2 all squared minus
Dialogue: 0,0:14:22.26,0:14:29.82,Default,,0000,0000,0000,,nine over 4 + 50 over\N3 and closed the big bracket.
Dialogue: 0,0:14:30.46,0:14:34.01,Default,,0000,0000,0000,,Now all we need to do now is put
Dialogue: 0,0:14:34.01,0:14:38.61,Default,,0000,0000,0000,,these together. And to do that\Nwe need a common denominator and
Dialogue: 0,0:14:38.61,0:14:41.74,Default,,0000,0000,0000,,the common denominator. Four and\Nthree is going to be 12.
Dialogue: 0,0:14:43.04,0:14:50.44,Default,,0000,0000,0000,,3 the Big Curly Bracket X minus\Nthree over 2 all squared minus
Dialogue: 0,0:14:50.44,0:14:57.26,Default,,0000,0000,0000,,over 12. We need to change the\Nnine over 4 into 12.
Dialogue: 0,0:14:58.41,0:15:02.81,Default,,0000,0000,0000,,3/4 gave us 12 so 3\Nnines give us 27.
Dialogue: 0,0:15:03.82,0:15:09.22,Default,,0000,0000,0000,,Plus we need to change the 50\Nover 3 into twelfths.
Dialogue: 0,0:15:09.82,0:15:15.83,Default,,0000,0000,0000,,4 * 3 is 12, so 4\N* 50 is 200.
Dialogue: 0,0:15:16.37,0:15:20.61,Default,,0000,0000,0000,,And now I've got a little bit of\Narithmetic to do. Let's just
Dialogue: 0,0:15:20.61,0:15:22.24,Default,,0000,0000,0000,,write back bracket down again.
Dialogue: 0,0:15:23.81,0:15:27.72,Default,,0000,0000,0000,,Equals 3. The Curly
Dialogue: 0,0:15:27.72,0:15:34.14,Default,,0000,0000,0000,,bracket. X minus three\Nover 2 all squared.
Dialogue: 0,0:15:35.24,0:15:38.91,Default,,0000,0000,0000,,Minus 27 over 12.
Dialogue: 0,0:15:40.06,0:15:47.74,Default,,0000,0000,0000,,Plus 200 over 12 and that's\Nthe calculation that we need to
Dialogue: 0,0:15:47.74,0:15:55.42,Default,,0000,0000,0000,,do here to simplify 3 Curly\Nbracket X minus three over 2
Dialogue: 0,0:15:55.42,0:16:01.86,Default,,0000,0000,0000,,or squared. Los all over 12\Nand we're going to take the 27
Dialogue: 0,0:16:01.86,0:16:07.46,Default,,0000,0000,0000,,away from the 200 is going to\Ngive us 173 and then we can
Dialogue: 0,0:16:07.46,0:16:09.06,Default,,0000,0000,0000,,close the bracket off.
Dialogue: 0,0:16:11.14,0:16:15.02,Default,,0000,0000,0000,,So despite the fact that the\Nnumbers were quite fearsome
Dialogue: 0,0:16:15.02,0:16:19.29,Default,,0000,0000,0000,,there, we've still ended up with\Na complete square, and we've
Dialogue: 0,0:16:19.29,0:16:23.56,Default,,0000,0000,0000,,automated the process so that\Nwhat we're doing is looking at
Dialogue: 0,0:16:23.56,0:16:28.21,Default,,0000,0000,0000,,the coefficient of X. First of\Nall, we check that what we've
Dialogue: 0,0:16:28.21,0:16:33.64,Default,,0000,0000,0000,,got the coefficient of X squared\Nis one. If it's not, we take out
Dialogue: 0,0:16:33.64,0:16:35.97,Default,,0000,0000,0000,,the coefficient of X squared as
Dialogue: 0,0:16:35.97,0:16:41.30,Default,,0000,0000,0000,,a factor. Next we check the\Ncoefficient of X and we take a
Dialogue: 0,0:16:41.30,0:16:45.59,Default,,0000,0000,0000,,half of it, and that's the\Nnumber that's going to go here
Dialogue: 0,0:16:45.59,0:16:46.67,Default,,0000,0000,0000,,inside the bracket.
Dialogue: 0,0:16:47.59,0:16:52.28,Default,,0000,0000,0000,,Then we must remember that we've\Ngot take off the square of that
Dialogue: 0,0:16:52.28,0:16:56.25,Default,,0000,0000,0000,,number is effectively we've\Nadded it back on, and then the
Dialogue: 0,0:16:56.25,0:16:57.70,Default,,0000,0000,0000,,rest is just arithmetic.
Dialogue: 0,0:16:59.70,0:17:03.67,Default,,0000,0000,0000,,So now we've developed this\Ntechnique of completing the
Dialogue: 0,0:17:03.67,0:17:08.52,Default,,0000,0000,0000,,square. Let's use it to solve\Nour original problem. If you
Dialogue: 0,0:17:08.52,0:17:14.69,Default,,0000,0000,0000,,remember we had X squared plus\N6X is equal to four and we chose
Dialogue: 0,0:17:14.69,0:17:19.99,Default,,0000,0000,0000,,to write that as X squared plus\N6X minus 4 equals 0.
Dialogue: 0,0:17:20.63,0:17:24.35,Default,,0000,0000,0000,,So first we need to check has it\Ngot a unit coefficient.
Dialogue: 0,0:17:24.89,0:17:29.96,Default,,0000,0000,0000,,And it has, so we don't need to\Ntake out a common factor. Now we
Dialogue: 0,0:17:29.96,0:17:34.35,Default,,0000,0000,0000,,look at the coefficient of X and\Nit's 6 and it's half that
Dialogue: 0,0:17:34.35,0:17:39.09,Default,,0000,0000,0000,,coefficient that we want. So we\Nneed 3. So this is going to be
Dialogue: 0,0:17:39.09,0:17:43.33,Default,,0000,0000,0000,,X. Plus three all\Nsquared.
Dialogue: 0,0:17:44.34,0:17:51.63,Default,,0000,0000,0000,,In doing that, we have added on\N3 squared, so we need to take
Dialogue: 0,0:17:51.63,0:17:53.72,Default,,0000,0000,0000,,off that 3 squared.
Dialogue: 0,0:17:54.26,0:17:57.17,Default,,0000,0000,0000,,And now we need to include the
Dialogue: 0,0:17:57.17,0:18:01.52,Default,,0000,0000,0000,,minus 4. So that we can\Nmaintain the quality of
Dialogue: 0,0:18:01.52,0:18:05.28,Default,,0000,0000,0000,,this xpression with this\None and it's equal to 0.
Dialogue: 0,0:18:06.29,0:18:10.15,Default,,0000,0000,0000,,So X plus three all squared.
Dialogue: 0,0:18:11.02,0:18:17.84,Default,,0000,0000,0000,,Minus 3 squared. That's minus\Nnine and minus 4 equals 0,
Dialogue: 0,0:18:17.84,0:18:25.28,Default,,0000,0000,0000,,so we can combine these X\Nplus three all squared minus 13
Dialogue: 0,0:18:25.28,0:18:31.52,Default,,0000,0000,0000,,equals not. At the 13 to each\Nside X plus three, all squared
Dialogue: 0,0:18:31.52,0:18:36.69,Default,,0000,0000,0000,,equals 13, and now we're in a\Nposition to take the square root
Dialogue: 0,0:18:36.69,0:18:41.87,Default,,0000,0000,0000,,of both sides. Because here on\Nthe left hand side we have a
Dialogue: 0,0:18:41.87,0:18:47.68,Default,,0000,0000,0000,,complete square. And so this is\NX +3 equals. Now 13 isn't a
Dialogue: 0,0:18:47.68,0:18:52.12,Default,,0000,0000,0000,,complete square. It's not a\Nsquare number, so we have to
Dialogue: 0,0:18:52.12,0:18:57.36,Default,,0000,0000,0000,,write it as square root of 13.\NOr remembering when we take a
Dialogue: 0,0:18:57.36,0:19:01.79,Default,,0000,0000,0000,,square root of a number it's\Nplus or minus Route 30.
Dialogue: 0,0:19:02.33,0:19:07.73,Default,,0000,0000,0000,,Now we take the three away from\Neach side and we end up with our
Dialogue: 0,0:19:07.73,0:19:12.77,Default,,0000,0000,0000,,two roots. So we take the three\Naway we have minus 3 + 13.
Dialogue: 0,0:19:13.38,0:19:18.33,Default,,0000,0000,0000,,Or minus 3 minus Route 30\Nand so that process of
Dialogue: 0,0:19:18.33,0:19:23.28,Default,,0000,0000,0000,,completing the square can be\Nused to help us solve a
Dialogue: 0,0:19:23.28,0:19:26.88,Default,,0000,0000,0000,,quadratic equation. But\Nthat's not the real issue
Dialogue: 0,0:19:26.88,0:19:30.93,Default,,0000,0000,0000,,here. You can see another\Nvideo on solving quadratic
Dialogue: 0,0:19:30.93,0:19:34.98,Default,,0000,0000,0000,,equations. The point is to\Nmaster this technique of
Dialogue: 0,0:19:34.98,0:19:36.33,Default,,0000,0000,0000,,completing the square.