[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.29,0:00:04.87,Default,,0000,0000,0000,,Completing the square is a\Nprocess that we make use of in a
Dialogue: 0,0:00:04.87,0:00:09.44,Default,,0000,0000,0000,,number of ways. First, we can\Nmake use of it to find maximum
Dialogue: 0,0:00:09.44,0:00:13.31,Default,,0000,0000,0000,,and minimum values of quadratic\Nfunctions, second we can make
Dialogue: 0,0:00:13.31,0:00:17.19,Default,,0000,0000,0000,,use of it to simplify or change\Nalgebraic expressions in order
Dialogue: 0,0:00:17.19,0:00:21.41,Default,,0000,0000,0000,,to be able to calculate the\Nvalue that they have. Third, we
Dialogue: 0,0:00:21.41,0:00:24.93,Default,,0000,0000,0000,,can use it for solving quadratic\Nequations. In this particular
Dialogue: 0,0:00:24.93,0:00:29.51,Default,,0000,0000,0000,,video, we're going to have a\Nlook at it for finding max- and
Dialogue: 0,0:00:29.51,0:00:32.21,Default,,0000,0000,0000,,min-imum values of functions, \Nquadratic functions.
Dialogue: 0,0:00:32.21,0:00:36.14,Default,,0000,0000,0000,,Let's begin by looking at a very specific\Nexample.
Dialogue: 0,0:00:36.14,0:00:39.20,Default,,0000,0000,0000,,Supposing we've got x squared,
Dialogue: 0,0:00:39.20,0:00:41.95,Default,,0000,0000,0000,,plus 5x,
Dialogue: 0,0:00:42.46,0:00:45.31,Default,,0000,0000,0000,,minus 2. Now.
Dialogue: 0,0:00:46.14,0:00:51.09,Default,,0000,0000,0000,,x squared, it's positive, so one\Nof the things that we do know is
Dialogue: 0,0:00:51.09,0:00:54.17,Default,,0000,0000,0000,,that if we were to sketch the\Ngraph of this function.
Dialogue: 0,0:00:54.98,0:00:57.29,Default,,0000,0000,0000,,It would look something perhaps
Dialogue: 0,0:00:57.29,0:01:02.73,Default,,0000,0000,0000,,like that. Question is where's\Nthis point down here?
Dialogue: 0,0:01:03.58,0:01:07.77,Default,,0000,0000,0000,,Where's the minimum value of\Nthis function? What value of x
Dialogue: 0,0:01:07.77,0:01:12.72,Default,,0000,0000,0000,,does it have? Does it actually\Ncome below the x-axis as I've
Dialogue: 0,0:01:12.72,0:01:17.68,Default,,0000,0000,0000,,drawn it, or does it come up\Nhere somewhere? At what value of X
Dialogue: 0,0:01:17.68,0:01:22.25,Default,,0000,0000,0000,,does that minimum value occur?\NWe could use calculus if we knew
Dialogue: 0,0:01:22.25,0:01:26.44,Default,,0000,0000,0000,,calculus, but sometimes we don't\Nknow calculus. We might not have
Dialogue: 0,0:01:26.44,0:01:27.58,Default,,0000,0000,0000,,reached it yet.
Dialogue: 0,0:01:28.42,0:01:31.91,Default,,0000,0000,0000,,At other occasions it might be\Nrather like using a sledgehammer
Dialogue: 0,0:01:31.91,0:01:36.66,Default,,0000,0000,0000,,to crack or not, so let's have a\Nlook at how we can deal with
Dialogue: 0,0:01:36.66,0:01:37.93,Default,,0000,0000,0000,,this kind of function.
Dialogue: 0,0:01:38.72,0:01:42.24,Default,,0000,0000,0000,,What we're going to do is\Na process known as
Dialogue: 0,0:01:42.38,0:01:50.11,Default,,0000,0000,0000,,"completing ... the ... square"
Dialogue: 0,0:01:51.97,0:01:55.27,Default,,0000,0000,0000,,OK, "completing the square",\Nwhat does that mean?
Dialogue: 0,0:01:55.45,0:02:00.14,Default,,0000,0000,0000,,Well, let's have a look at something\Nthat is a "complete square".
Dialogue: 0,0:02:00.30,0:02:05.04,Default,,0000,0000,0000,,That is, an {\i1}exact{\i0} square.
Dialogue: 0,0:02:06.11,0:02:12.66,Default,,0000,0000,0000,,So that's a complete and exact\Nsquare. If we multiply out the
Dialogue: 0,0:02:12.66,0:02:20.31,Default,,0000,0000,0000,,brackets, x plus a times by x\Nplus a, what we end up with
Dialogue: 0,0:02:20.31,0:02:21.94,Default,,0000,0000,0000,,is x squared...
Dialogue: 0,0:02:22.69,0:02:25.07,Default,,0000,0000,0000,,that's x times by x...
Dialogue: 0,0:02:26.32,0:02:33.49,Default,,0000,0000,0000,,a times by x, and of course\Nx times by a, so that gives
Dialogue: 0,0:02:33.49,0:02:38.34,Default,,0000,0000,0000,,us 2ax, and then finally a\Ntimes by a...
Dialogue: 0,0:02:38.39,0:02:40.66,Default,,0000,0000,0000,,and that gives us a squared.
Dialogue: 0,0:02:40.66,0:02:46.29,Default,,0000,0000,0000,,So this expression is\Na complete square, a complete
Dialogue: 0,0:02:46.29,0:02:51.41,Default,,0000,0000,0000,,and exact square. Because it's "x\Nplus a" all squared.
Dialogue: 0,0:02:52.41,0:02:57.37,Default,,0000,0000,0000,,Similarly, we can have "x minus a"\Nall squared.
Dialogue: 0,0:02:58.34,0:03:00.64,Default,,0000,0000,0000,,And if we\Nmultiply out, these brackets
Dialogue: 0,0:03:00.78,0:03:06.80,Default,,0000,0000,0000,,we will end up with the same\Nresult, {\i1}except{\i0}, we will have
Dialogue: 0,0:03:06.80,0:03:12.56,Default,,0000,0000,0000,,{\i1}minus{\i0} 2ax plus a squared. And\Nagain this is a complete
Dialogue: 0,0:03:12.56,0:03:17.28,Default,,0000,0000,0000,,square an exact square\Nbecause it's equal to x minus a...
Dialogue: 0,0:03:17.28,0:03:19.37,Default,,0000,0000,0000,,all squared.
Dialogue: 0,0:03:20.79,0:03:23.88,Default,,0000,0000,0000,,So,... we go back to this.
Dialogue: 0,0:03:24.45,0:03:28.86,Default,,0000,0000,0000,,Expression here x squared, plus\N5x, minus two and what we're
Dialogue: 0,0:03:28.86,0:03:33.67,Default,,0000,0000,0000,,going to do is complete the\Nsquare. In other words we're going to
Dialogue: 0,0:03:33.67,0:03:38.89,Default,,0000,0000,0000,,try and make it look like this.\NWe're going to try and {\i1}complete{\i0} it.
Dialogue: 0,0:03:38.89,0:03:44.10,Default,,0000,0000,0000,,Make it up so it's a full\Nsquare. In order to do that,
Dialogue: 0,0:03:44.10,0:03:48.11,Default,,0000,0000,0000,,what we're going to do is\Ncompare that expression directly
Dialogue: 0,0:03:48.11,0:03:49.31,Default,,0000,0000,0000,,with that one.
Dialogue: 0,0:03:50.08,0:03:55.96,Default,,0000,0000,0000,,And we've chosen this expression\Nhere because that's a plus sign
Dialogue: 0,0:03:55.96,0:04:01.85,Default,,0000,0000,0000,,plus 5x, and that's a plus sign\Nthere plus 2ax.
Dialogue: 0,0:04:01.85,0:04:05.48,Default,,0000,0000,0000,,So.
Dialogue: 0,0:04:05.50,0:04:11.79,Default,,0000,0000,0000,,x squared, plus\N5x, minus 2.
Dialogue: 0,0:04:12.52,0:04:22.14,Default,,0000,0000,0000,,And we have x squared\Nplus 2ax plus a squared
Dialogue: 0,0:04:22.17,0:04:24.32,Default,,0000,0000,0000,,These two match up
Dialogue: 0,0:04:24.73,0:04:29.85,Default,,0000,0000,0000,,Somehow we've got\Nto match these two up.
Dialogue: 0,0:04:29.85,0:04:31.62,Default,,0000,0000,0000,,Well,... the x's are the same.
Dialogue: 0,0:04:32.26,0:04:38.100,Default,,0000,0000,0000,,So the 5 and the 2a have got\Nto be the same and that would
Dialogue: 0,0:04:38.100,0:04:44.38,Default,,0000,0000,0000,,suggest to us that a has got to\Nbe 5 / 2.
Dialogue: 0,0:04:44.98,0:04:50.84,Default,,0000,0000,0000,,So that x squared plus\N5x minus 2...
Dialogue: 0,0:04:52.51,0:04:59.35,Default,,0000,0000,0000,,becomes x squared plus 5x...
Dialogue: 0,0:04:59.63,0:05:02.15,Default,,0000,0000,0000,,now... plus a squared and
Dialogue: 0,0:05:02.69,0:05:06.31,Default,,0000,0000,0000,,now we decided that 5 was\Nto be equal to 2a
Dialogue: 0,0:05:06.31,0:05:12.52,Default,,0000,0000,0000,,and so a was equal\Nto 5 over 2.
Dialogue: 0,0:05:13.09,0:05:20.21,Default,,0000,0000,0000,,So to complete the square, we've\Ngot to add on 5 over 2...
Dialogue: 0,0:05:20.21,0:05:21.86,Default,,0000,0000,0000,,and square it.
Dialogue: 0,0:05:24.40,0:05:26.20,Default,,0000,0000,0000,,But that isn't equal to that.
Dialogue: 0,0:05:28.36,0:05:32.65,Default,,0000,0000,0000,,It's equal, this is equal\Nto that, but not to that
Dialogue: 0,0:05:32.65,0:05:36.55,Default,,0000,0000,0000,,well. Clearly we need to\Nput the minus two on.
Dialogue: 0,0:05:38.60,0:05:42.85,Default,,0000,0000,0000,,But then it's still not equal,\Nbecause here we've added on
Dialogue: 0,0:05:42.85,0:05:47.86,Default,,0000,0000,0000,,something extra 5 over 2 [squared]. So\Nwe've got to take off that five
Dialogue: 0,0:05:47.86,0:05:51.72,Default,,0000,0000,0000,,over 2 all squared. We've got to\Ntake that away.
Dialogue: 0,0:05:53.41,0:05:56.28,Default,,0000,0000,0000,,Now let's look at this bit.
Dialogue: 0,0:05:57.14,0:06:02.21,Default,,0000,0000,0000,,This is an exact square. It's\Nthat expression there.
Dialogue: 0,0:06:03.06,0:06:10.56,Default,,0000,0000,0000,,No, this began life as x\Nplus a all squared, so this
Dialogue: 0,0:06:10.56,0:06:18.06,Default,,0000,0000,0000,,bit has got to be the\Nsame, x plus (5 over 2) all
Dialogue: 0,0:06:18.06,0:06:25.08,Default,,0000,0000,0000,,squared. And now we can\Nplay with this. We've got minus 2
Dialogue: 0,0:06:25.08,0:06:31.96,Default,,0000,0000,0000,,minus 25 over 4. We\Ncan combine that so we have
Dialogue: 0,0:06:31.96,0:06:35.39,Default,,0000,0000,0000,,x plus (5 over 2) all squared...
Dialogue: 0,0:06:36.16,0:06:37.39,Default,,0000,0000,0000,,minus...
Dialogue: 0,0:06:38.60,0:06:43.24,Default,,0000,0000,0000,,Now we're taking away two, so\Nin terms of quarters, that's
Dialogue: 0,0:06:43.24,0:06:47.46,Default,,0000,0000,0000,,8 quarters were taking\Naway, and we're taking away 25
Dialogue: 0,0:06:47.46,0:06:50.84,Default,,0000,0000,0000,,quarters as well, so\Naltogether, that's 33
Dialogue: 0,0:06:50.84,0:06:52.95,Default,,0000,0000,0000,,quarters that we're taking\Naway.
Dialogue: 0,0:06:54.40,0:06:59.07,Default,,0000,0000,0000,,Now let's have a look at this\Nexpression... x squared plus 5x minus 2.
Dialogue: 0,0:06:59.07,0:07:03.35,Default,,0000,0000,0000,,Remember what we were\Nasking was "what's its minimum value?"
Dialogue: 0,0:07:03.35,0:07:06.82,Default,,0000,0000,0000,,Its graph looked like that.\NWe were interested in...
Dialogue: 0,0:07:06.82,0:07:08.00,Default,,0000,0000,0000,,"where's this point?"
Dialogue: 0,0:07:08.13,0:07:09.80,Default,,0000,0000,0000,,"where is the lowest point?"
Dialogue: 0,0:07:10.17,0:07:11.73,Default,,0000,0000,0000,,"what's the x-coordinate?"
Dialogue: 0,0:07:11.90,0:07:13.87,Default,,0000,0000,0000,,"and what's the y-coordinate?"
Dialogue: 0,0:07:14.53,0:07:16.99,Default,,0000,0000,0000,,Let's have a look at this\Nexpression here.
Dialogue: 0,0:07:17.11,0:07:21.90,Default,,0000,0000,0000,,This is a square.\NA square is always
Dialogue: 0,0:07:21.90,0:07:26.72,Default,,0000,0000,0000,,positive {\i1}unless{\i0} it's equal to 0,\Nso its lowest value
Dialogue: 0,0:07:26.93,0:07:30.64,Default,,0000,0000,0000,,that this expression [can take] is 0.
Dialogue: 0,0:07:30.78,0:07:34.55,Default,,0000,0000,0000,,So the lowest value of\Nthe whole expression...
Dialogue: 0,0:07:34.59,0:07:38.69,Default,,0000,0000,0000,,is {\i1}that{\i0} "minus 33 over 4".
Dialogue: 0,0:07:38.93,0:07:43.07,Default,,0000,0000,0000,,So therefore we can say \Nthat the minimum value...
Dialogue: 0,0:07:48.59,0:07:55.43,Default,,0000,0000,0000,,of x squared, plus 5x, minus\N2 equals... minus 33 over 4.
Dialogue: 0,0:07:55.95,0:07:58.56,Default,,0000,0000,0000,,And we need to be able\Nto say {\i1}when{\i0}
Dialogue: 0,0:07:58.88,0:08:00.61,Default,,0000,0000,0000,,"what's the x-value there?"
Dialogue: 0,0:08:00.86,0:08:04.93,Default,,0000,0000,0000,,well, it occurs when this bracket\Nis at its lowest value.
Dialogue: 0,0:08:05.19,0:08:11.14,Default,,0000,0000,0000,,When this bracket is at\N0. In other words, when x equals...
Dialogue: 0,0:08:11.14,0:08:14.46,Default,,0000,0000,0000,,minus 5 over 2.
Dialogue: 0,0:08:16.55,0:08:23.38,Default,,0000,0000,0000,,So we found the minimum\Nvalue and exactly {\i1}when{\i0} it happens.
Dialogue: 0,0:08:25.85,0:08:31.90,Default,,0000,0000,0000,,Let's take a second example. Our\Nquadratic function this time, f of x,
Dialogue: 0,0:08:31.90,0:08:37.40,Default,,0000,0000,0000,,is x squared, minus 6\Nx, minus 12.
Dialogue: 0,0:08:38.21,0:08:45.25,Default,,0000,0000,0000,,We've got a minus sign in here, so\Nlet's line this up with the
Dialogue: 0,0:08:45.25,0:08:50.54,Default,,0000,0000,0000,,complete square: x squared, minus\N2ax, plus a squared.
Dialogue: 0,0:08:51.21,0:08:56.01,Default,,0000,0000,0000,,The x squared terms are the same,\Nand we want these two to be the
Dialogue: 0,0:08:56.01,0:09:00.49,Default,,0000,0000,0000,,same as well. That clearly means\Nthat 2a has got to be the same
Dialogue: 0,0:09:00.49,0:09:03.05,Default,,0000,0000,0000,,as 6, so a has got to be 3.
Dialogue: 0,0:09:03.61,0:09:09.79,Default,,0000,0000,0000,,So f(x) is equal\Nto x squared, minus 6x,
Dialogue: 0,0:09:09.79,0:09:11.56,Default,,0000,0000,0000,,plus...
Dialogue: 0,0:09:11.56,0:09:17.34,Default,,0000,0000,0000,,a squared (which is 3 squared),\Nminus 12, and now we added on
Dialogue: 0,0:09:17.34,0:09:22.46,Default,,0000,0000,0000,,3 squared. So we've got to take\Nthe 3 squared away in order
Dialogue: 0,0:09:22.46,0:09:27.59,Default,,0000,0000,0000,,to make it equal. To keep the\Nvalue of the original expression
Dialogue: 0,0:09:27.59,0:09:29.29,Default,,0000,0000,0000,,that we started with.
Dialogue: 0,0:09:29.93,0:09:37.77,Default,,0000,0000,0000,,We can now identify this as\Nbeing (x minus 3) all squared.
Dialogue: 0,0:09:39.06,0:09:41.17,Default,,0000,0000,0000,,And these numbers at the end...
Dialogue: 0,0:09:41.75,0:09:47.92,Default,,0000,0000,0000,,minus 12 minus 9, altogether\Ngives us minus 21.
Dialogue: 0,0:09:49.02,0:09:54.61,Default,,0000,0000,0000,,Again, we can say does it have a\Nmaximum value or a minimum value?
Dialogue: 0,0:09:54.61,0:09:57.73,Default,,0000,0000,0000,,Well what we know that we began\Nwith a positive x squared term,
Dialogue: 0,0:09:57.73,0:10:01.82,Default,,0000,0000,0000,,so the shape of the graph\Nhas got to be like that. So we
Dialogue: 0,0:10:01.82,0:10:03.74,Default,,0000,0000,0000,,know that we're looking for a
Dialogue: 0,0:10:03.74,0:10:08.22,Default,,0000,0000,0000,,minimum value. We know that that\Nminimum value will occur when
Dialogue: 0,0:10:08.22,0:10:13.61,Default,,0000,0000,0000,,this bit is 0 because it's a\Nsquare, it's least value is
Dialogue: 0,0:10:13.61,0:10:21.34,Default,,0000,0000,0000,,going to be 0, so therefore we\Ncan say the minimum value.
Dialogue: 0,0:10:22.09,0:10:25.100,Default,,0000,0000,0000,,of our quadratic function f of x
Dialogue: 0,0:10:25.100,0:10:31.86,Default,,0000,0000,0000,,is minus 21, [occurring] {\i1}when{\i0}...
Dialogue: 0,0:10:32.87,0:10:38.95,Default,,0000,0000,0000,,this bit is 0. In other words,\Nwhen x equals 3.
Dialogue: 0,0:10:40.65,0:10:46.18,Default,,0000,0000,0000,,The two examples we've taken so far\Nhave both had a positive x squared
Dialogue: 0,0:10:46.18,0:10:51.71,Default,,0000,0000,0000,,and a unit coefficient\Nof x squared, in other words 1 x squared.
Dialogue: 0,0:10:51.71,0:10:57.14,Default,,0000,0000,0000,,We'll now look at an example \Nwhere we've got a number here
Dialogue: 0,0:10:57.14,0:10:59.91,Default,,0000,0000,0000,,in front of the x squared.
Dialogue: 0,0:10:59.92,0:11:04.16,Default,,0000,0000,0000,,So the example\Nthat will take.
Dialogue: 0,0:11:05.98,0:11:11.29,Default,,0000,0000,0000,,f of x equals 2x squared,
Dialogue: 0,0:11:11.57,0:11:16.55,Default,,0000,0000,0000,,minus 6x, plus one.
Dialogue: 0,0:11:17.26,0:11:22.85,Default,,0000,0000,0000,,Our first step is to take out\Nthat 2 as a factor.
Dialogue: 0,0:11:23.80,0:11:27.87,Default,,0000,0000,0000,,2, brackets x squared,
Dialogue: 0,0:11:28.48,0:11:31.02,Default,,0000,0000,0000,,minus 3x,...
Dialogue: 0,0:11:31.63,0:11:37.37,Default,,0000,0000,0000,,we've got to take\Nthe 2 out of this as well, so
Dialogue: 0,0:11:37.37,0:11:38.70,Default,,0000,0000,0000,,that's a half.
Dialogue: 0,0:11:39.37,0:11:45.09,Default,,0000,0000,0000,,And now we do the same as we've\Ndone before with this bracket here.
Dialogue: 0,0:11:45.77,0:11:52.65,Default,,0000,0000,0000,,We line this one up with x squared,\Nminus 2ax, plus a squared.
Dialogue: 0,0:11:53.38,0:11:59.99,Default,,0000,0000,0000,,When making these two terms the\Nsame 3 has to be the same as 2a,
Dialogue: 0,0:11:59.99,0:12:05.65,Default,,0000,0000,0000,,and so 3 over 2 has to\Nbe equal to a.
Dialogue: 0,0:12:06.77,0:12:13.74,Default,,0000,0000,0000,,So our function f of x is going\Nto be equal to 2 times...
Dialogue: 0,0:12:14.53,0:12:18.44,Default,,0000,0000,0000,,x squared, minus 3x,...
Dialogue: 0,0:12:19.17,0:12:27.27,Default,,0000,0000,0000,,now we want plus a squared,\Nso that's plus (3 over 2) all squared
Dialogue: 0,0:12:28.26,0:12:31.74,Default,,0000,0000,0000,,Plus the half that\Nwas there originally and now
Dialogue: 0,0:12:31.74,0:12:35.91,Default,,0000,0000,0000,,we've added on this, so we've\Ngot to take it away,...
Dialogue: 0,0:12:36.14,0:12:42.42,Default,,0000,0000,0000,,(3 over 2) all squared. And finally we\Nopened a bracket, so we must
Dialogue: 0,0:12:42.42,0:12:44.55,Default,,0000,0000,0000,,close it at the end.
Dialogue: 0,0:12:45.74,0:12:53.02,Default,,0000,0000,0000,,Equals... 2, bracket,... now this\Nis going to be our complete square
Dialogue: 0,0:12:53.02,0:12:59.40,Default,,0000,0000,0000,,(x minus 3 over 2) all squared.
Dialogue: 0,0:12:59.93,0:13:05.37,Default,,0000,0000,0000,,And then here we've got some\Ncalculation to do. We've plus a half,
Dialogue: 0,0:13:05.65,0:13:12.55,Default,,0000,0000,0000,,take away (3 over 2) squared, \Nso that's plus 1/2
Dialogue: 0,0:13:12.55,0:13:14.96,Default,,0000,0000,0000,,take away 9 over 4.
Dialogue: 0,0:13:16.62,0:13:21.23,Default,,0000,0000,0000,,The front bit is going to stay the same
Dialogue: 0,0:13:24.51,0:13:27.23,Default,,0000,0000,0000,,And now we can juggle with\Nthese fractions. At the end,
Dialogue: 0,0:13:27.46,0:13:32.02,Default,,0000,0000,0000,,we've got plus 1/2 take away 9\Nquarters or 1/2 is 2 quarters,
Dialogue: 0,0:13:32.02,0:13:36.55,Default,,0000,0000,0000,,so if we're taking\Naway, nine quarters must be
Dialogue: 0,0:13:36.99,0:13:40.10,Default,,0000,0000,0000,,ultimately taking away 7 quarters.
Dialogue: 0,0:13:40.36,0:13:44.93,Default,,0000,0000,0000,,So again, what's the\Nminimum value of this function?
Dialogue: 0,0:13:44.93,0:13:49.04,Default,,0000,0000,0000,,It had a positive 2 in front\Nof the x squared, so again, it
Dialogue: 0,0:13:49.04,0:13:52.40,Default,,0000,0000,0000,,looks like that. And again,\Nwe're asking the question,
Dialogue: 0,0:13:52.40,0:13:56.50,Default,,0000,0000,0000,,"what's this point down here?"\NWhat's the lowest point and that
Dialogue: 0,0:13:56.50,0:13:59.48,Default,,0000,0000,0000,,lowest point must occur when\Nthis is 0.
Dialogue: 0,0:14:00.34,0:14:02.35,Default,,0000,0000,0000,,So the min ...
Dialogue: 0,0:14:02.89,0:14:10.83,Default,,0000,0000,0000,,value of f of x must be\Nequal to... now that's going to be 0
Dialogue: 0,0:14:10.83,0:14:18.27,Default,,0000,0000,0000,,But we're still multiplying\Nby the 2, so it's 2 times
Dialogue: 0,0:14:18.27,0:14:24.72,Default,,0000,0000,0000,,minus 7 over 4. That's minus\N14 over 4, which reduces to
Dialogue: 0,0:14:24.72,0:14:27.41,Default,,0000,0000,0000,,minus 7 over 2. When?
Dialogue: 0,0:14:28.10,0:14:34.76,Default,,0000,0000,0000,,And that will happen when this\Nis zero. In other words, when x
Dialogue: 0,0:14:34.76,0:14:36.80,Default,,0000,0000,0000,,equals 3 over 2.
Dialogue: 0,0:14:38.17,0:14:43.96,Default,,0000,0000,0000,,So a minimum value of minus\N7 over 2 when x equals 3 over 2.
Dialogue: 0,0:14:43.96,0:14:49.30,Default,,0000,0000,0000,,Let's take one final\Nexample and this time when the
Dialogue: 0,0:14:49.30,0:14:51.52,Default,,0000,0000,0000,,coefficient of x squared is
Dialogue: 0,0:14:51.52,0:14:53.99,Default,,0000,0000,0000,,actually negative.
Dialogue: 0,0:14:54.11,0:14:57.99,Default,,0000,0000,0000,,So for this will take our quadratic function\Nto be f of x
Dialogue: 0,0:14:58.60,0:15:02.08,Default,,0000,0000,0000,,equals... 3 plus
Dialogue: 0,0:15:02.08,0:15:08.51,Default,,0000,0000,0000,,8x minus\N2(x squared).
Dialogue: 0,0:15:09.52,0:15:16.00,Default,,0000,0000,0000,,We operate in just the same way\Nas we did before. We take out
Dialogue: 0,0:15:16.00,0:15:21.10,Default,,0000,0000,0000,,the factor that is multiplying\Nthe x squared and on this
Dialogue: 0,0:15:21.10,0:15:24.34,Default,,0000,0000,0000,,occasion it's minus 2. \NThe "- 2" comes out.
Dialogue: 0,0:15:24.34,0:15:31.80,Default,,0000,0000,0000,,Times by x squared, we\Ntake a minus 2 out of the 8x,
Dialogue: 0,0:15:31.80,0:15:39.08,Default,,0000,0000,0000,,that leaves us minus 4x and the\Nminus 2 out of the 3 is a
Dialogue: 0,0:15:39.08,0:15:42.96,Default,,0000,0000,0000,,factor which gives us minus\Nthree over 2.
Dialogue: 0,0:15:44.76,0:15:50.87,Default,,0000,0000,0000,,We line this one up with X\Nsquared minus 2X plus A squared.
Dialogue: 0,0:15:50.87,0:15:57.92,Default,,0000,0000,0000,,Those two are the same. We want\Nthese two to be the same. 2A is
Dialogue: 0,0:15:57.92,0:16:03.56,Default,,0000,0000,0000,,equal to four, so a has got to\Nbe equal to two.
Dialogue: 0,0:16:04.12,0:16:10.32,Default,,0000,0000,0000,,So our F of X is going\Nto be minus 2.
Dialogue: 0,0:16:11.28,0:16:17.11,Default,,0000,0000,0000,,X squared minus 4X plus A\Nsquared, so that's +2 squared
Dialogue: 0,0:16:17.11,0:16:23.47,Default,,0000,0000,0000,,minus the original 3 over 2,\Nbut we've added on a 2
Dialogue: 0,0:16:23.47,0:16:30.36,Default,,0000,0000,0000,,squared, so we need to take it\Naway again to keep the balance
Dialogue: 0,0:16:30.36,0:16:32.48,Default,,0000,0000,0000,,to keep the equality.
Dialogue: 0,0:16:33.82,0:16:40.40,Default,,0000,0000,0000,,Minus two, this is now our\Ncomplete square, so that's X
Dialogue: 0,0:16:40.40,0:16:47.57,Default,,0000,0000,0000,,minus two all squared. And here\Nwe've got minus three over 2
Dialogue: 0,0:16:47.57,0:16:54.44,Default,,0000,0000,0000,,- 4. Well, let's have it all\Nover too. So minus four is minus
Dialogue: 0,0:16:54.44,0:16:59.14,Default,,0000,0000,0000,,8 over 2, so altogether we've\Ngot minus 11 over 2.
Dialogue: 0,0:17:00.91,0:17:06.19,Default,,0000,0000,0000,,And we can look at this. We\Ncan see that when this is
Dialogue: 0,0:17:06.19,0:17:10.65,Default,,0000,0000,0000,,zero, we've got. In this case\Na maximum value, because this
Dialogue: 0,0:17:10.65,0:17:15.53,Default,,0000,0000,0000,,is a negative X squared term.\NSo we know that we're looking
Dialogue: 0,0:17:15.53,0:17:20.40,Default,,0000,0000,0000,,for a graph like that. So it's\Nthis point that we're looking
Dialogue: 0,0:17:20.40,0:17:24.05,Default,,0000,0000,0000,,for the maximum point, and so\Ntherefore maximum value.
Dialogue: 0,0:17:26.30,0:17:28.31,Default,,0000,0000,0000,,Solve F of X.
Dialogue: 0,0:17:28.81,0:17:33.83,Default,,0000,0000,0000,,Will occur when this square term\Nis equal to 0 'cause the square
Dialogue: 0,0:17:33.83,0:17:36.53,Default,,0000,0000,0000,,term can never be less than 0.
Dialogue: 0,0:17:37.42,0:17:40.58,Default,,0000,0000,0000,,And so we have minus two times.
Dialogue: 0,0:17:41.08,0:17:46.78,Default,,0000,0000,0000,,Minus 11. And altogether that\Ngives us plus 11.
Dialogue: 0,0:17:47.30,0:17:51.24,Default,,0000,0000,0000,,And it will occur when this.
Dialogue: 0,0:17:51.75,0:17:58.93,Default,,0000,0000,0000,,Is equal to 0. In other\Nwords, when X equals 2.