[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.44,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.44,0:00:02.93,Default,,0000,0000,0000,,In this video, I'm going to show\Nyou a technique called
Dialogue: 0,0:00:02.93,0:00:09.31,Default,,0000,0000,0000,,completing the square.
Dialogue: 0,0:00:09.31,0:00:14.49,Default,,0000,0000,0000,,And what's neat about this is\Nthat this will work for any
Dialogue: 0,0:00:14.49,0:00:18.21,Default,,0000,0000,0000,,quadratic equation, and it's\Nactually the basis for the
Dialogue: 0,0:00:18.21,0:00:18.75,Default,,0000,0000,0000,,quadratic formula.
Dialogue: 0,0:00:18.75,0:00:21.99,Default,,0000,0000,0000,,And in the next video or the\Nvideo after that I'll prove
Dialogue: 0,0:00:21.99,0:00:25.63,Default,,0000,0000,0000,,the quadratic formula using\Ncompleting the square.
Dialogue: 0,0:00:25.63,0:00:28.45,Default,,0000,0000,0000,,But before we do that, we\Nneed to understand even
Dialogue: 0,0:00:28.45,0:00:29.47,Default,,0000,0000,0000,,what it's all about.
Dialogue: 0,0:00:29.47,0:00:32.07,Default,,0000,0000,0000,,And it really just builds off\Nof what we did in the last
Dialogue: 0,0:00:32.07,0:00:33.88,Default,,0000,0000,0000,,video, where we solved\Nquadratics
Dialogue: 0,0:00:33.88,0:00:36.13,Default,,0000,0000,0000,,using perfect squares.
Dialogue: 0,0:00:36.13,0:00:39.90,Default,,0000,0000,0000,,So let's say I have the\Nquadratic equation x squared
Dialogue: 0,0:00:39.90,0:00:44.88,Default,,0000,0000,0000,,minus 4x is equal to 5.
Dialogue: 0,0:00:44.88,0:00:47.49,Default,,0000,0000,0000,,And I put this big space\Nhere for a reason.
Dialogue: 0,0:00:47.49,0:00:49.68,Default,,0000,0000,0000,,In the last video, we saw\Nthat these can be pretty
Dialogue: 0,0:00:49.68,0:00:53.20,Default,,0000,0000,0000,,straightforward to solve if\Nthe left-hand side is a
Dialogue: 0,0:00:53.20,0:00:56.50,Default,,0000,0000,0000,,perfect square.
Dialogue: 0,0:00:56.50,0:00:59.05,Default,,0000,0000,0000,,You see, completing the square\Nis all about making the
Dialogue: 0,0:00:59.05,0:01:01.90,Default,,0000,0000,0000,,quadratic equation into a\Nperfect square, engineering
Dialogue: 0,0:01:01.90,0:01:05.19,Default,,0000,0000,0000,,it, adding and subtracting from\Nboth sides so it becomes
Dialogue: 0,0:01:05.19,0:01:05.97,Default,,0000,0000,0000,,a perfect square.
Dialogue: 0,0:01:05.97,0:01:07.71,Default,,0000,0000,0000,,So how can we do that?
Dialogue: 0,0:01:07.71,0:01:10.13,Default,,0000,0000,0000,,Well, in order for this\Nleft-hand side to be a perfect
Dialogue: 0,0:01:10.13,0:01:12.99,Default,,0000,0000,0000,,square, there has to be\Nsome number here.
Dialogue: 0,0:01:12.99,0:01:17.51,Default,,0000,0000,0000,,There has to be some number here\Nthat if I have my number
Dialogue: 0,0:01:17.51,0:01:20.91,Default,,0000,0000,0000,,squared I get that number, and\Nthen if I have two times my
Dialogue: 0,0:01:20.91,0:01:22.89,Default,,0000,0000,0000,,number I get negative 4.
Dialogue: 0,0:01:22.89,0:01:24.75,Default,,0000,0000,0000,,Remember that, and I\Nthink it'll become
Dialogue: 0,0:01:24.75,0:01:27.70,Default,,0000,0000,0000,,clear with a few examples.
Dialogue: 0,0:01:27.70,0:01:35.23,Default,,0000,0000,0000,,I want x squared minus 4x plus\Nsomething to be equal to x
Dialogue: 0,0:01:35.23,0:01:37.74,Default,,0000,0000,0000,,minus a squared.
Dialogue: 0,0:01:37.74,0:01:41.01,Default,,0000,0000,0000,,We don't know what a\Nis just yet, but we
Dialogue: 0,0:01:41.01,0:01:42.11,Default,,0000,0000,0000,,know a couple of things.
Dialogue: 0,0:01:42.11,0:01:46.18,Default,,0000,0000,0000,,When I square things-- so this\Nis going to be x squared minus
Dialogue: 0,0:01:46.18,0:01:49.33,Default,,0000,0000,0000,,2a plus a squared.
Dialogue: 0,0:01:49.33,0:01:53.64,Default,,0000,0000,0000,,So if you look at this pattern\Nright here, that has to be--
Dialogue: 0,0:01:53.64,0:01:59.88,Default,,0000,0000,0000,,sorry, x squared minus 2ax--\Nthis right here has to be 2ax.
Dialogue: 0,0:01:59.88,0:02:03.53,Default,,0000,0000,0000,,And this right here would\Nhave to be a squared.
Dialogue: 0,0:02:03.53,0:02:07.69,Default,,0000,0000,0000,,So this number, a is going to\Nbe half of negative 4, a has
Dialogue: 0,0:02:07.69,0:02:10.37,Default,,0000,0000,0000,,to be negative 2, right?
Dialogue: 0,0:02:10.37,0:02:13.57,Default,,0000,0000,0000,,Because 2 times a is going\Nto be negative 4.
Dialogue: 0,0:02:13.57,0:02:18.33,Default,,0000,0000,0000,,a is negative 2, and if a is\Nnegative 2, what is a squared?
Dialogue: 0,0:02:18.33,0:02:21.55,Default,,0000,0000,0000,,Well, then a squared is going\Nto be positive 4.
Dialogue: 0,0:02:21.55,0:02:24.22,Default,,0000,0000,0000,,And this might look all\Ncomplicated to you right now,
Dialogue: 0,0:02:24.22,0:02:25.91,Default,,0000,0000,0000,,but I'm showing you\Nthe rationale.
Dialogue: 0,0:02:25.91,0:02:29.08,Default,,0000,0000,0000,,You literally just look at this\Ncoefficient right here,
Dialogue: 0,0:02:29.08,0:02:32.67,Default,,0000,0000,0000,,and you say, OK, well what's\Nhalf of that coefficient?
Dialogue: 0,0:02:32.67,0:02:35.92,Default,,0000,0000,0000,,Well, half of that coefficient\Nis negative 2.
Dialogue: 0,0:02:35.92,0:02:40.23,Default,,0000,0000,0000,,So we could say a is equal to\Nnegative 2-- same idea there--
Dialogue: 0,0:02:40.23,0:02:41.72,Default,,0000,0000,0000,,and then you square it.
Dialogue: 0,0:02:41.72,0:02:44.10,Default,,0000,0000,0000,,You square a, you\Nget positive 4.
Dialogue: 0,0:02:44.10,0:02:46.54,Default,,0000,0000,0000,,So we add positive 4 here.
Dialogue: 0,0:02:46.54,0:02:47.63,Default,,0000,0000,0000,,Add a 4.
Dialogue: 0,0:02:47.63,0:02:50.99,Default,,0000,0000,0000,,Now, from the very first\Nequation we ever did, you
Dialogue: 0,0:02:50.99,0:02:55.24,Default,,0000,0000,0000,,should know that you can never\Ndo something to just one side
Dialogue: 0,0:02:55.24,0:02:55.90,Default,,0000,0000,0000,,of the equation.
Dialogue: 0,0:02:55.90,0:02:58.70,Default,,0000,0000,0000,,You can't add 4 to just one\Nside of the equation.
Dialogue: 0,0:02:58.70,0:03:02.71,Default,,0000,0000,0000,,If x squared minus 4x was equal\Nto 5, then when I add 4
Dialogue: 0,0:03:02.71,0:03:04.72,Default,,0000,0000,0000,,it's not going to be\Nequal to 5 anymore.
Dialogue: 0,0:03:04.72,0:03:07.95,Default,,0000,0000,0000,,It's going to be equal\Nto 5 plus 4.
Dialogue: 0,0:03:07.95,0:03:11.43,Default,,0000,0000,0000,,We added 4 on the left-hand side\Nbecause we wanted this to
Dialogue: 0,0:03:11.43,0:03:12.44,Default,,0000,0000,0000,,be a perfect square.
Dialogue: 0,0:03:12.44,0:03:15.21,Default,,0000,0000,0000,,But if you add something to the\Nleft-hand side, you've got
Dialogue: 0,0:03:15.21,0:03:17.32,Default,,0000,0000,0000,,to add it to the right-hand\Nside.
Dialogue: 0,0:03:17.32,0:03:20.63,Default,,0000,0000,0000,,And now, we've gotten ourselves\Nto a problem that's
Dialogue: 0,0:03:20.63,0:03:23.41,Default,,0000,0000,0000,,just like the problems we\Ndid in the last video.
Dialogue: 0,0:03:23.41,0:03:25.96,Default,,0000,0000,0000,,What is this left-hand side?
Dialogue: 0,0:03:25.96,0:03:27.00,Default,,0000,0000,0000,,Let me rewrite the\Nwhole thing.
Dialogue: 0,0:03:27.00,0:03:33.02,Default,,0000,0000,0000,,We have x squared minus 4x\Nplus 4 is equal to 9 now.
Dialogue: 0,0:03:33.02,0:03:35.38,Default,,0000,0000,0000,,All we did is add 4 to both\Nsides of the equation.
Dialogue: 0,0:03:35.38,0:03:39.07,Default,,0000,0000,0000,,But we added 4 on purpose so\Nthat this left-hand side
Dialogue: 0,0:03:39.07,0:03:41.08,Default,,0000,0000,0000,,becomes a perfect square.
Dialogue: 0,0:03:41.08,0:03:41.76,Default,,0000,0000,0000,,Now what is this?
Dialogue: 0,0:03:41.76,0:03:45.34,Default,,0000,0000,0000,,What number when I multiply it\Nby itself is equal to 4 and
Dialogue: 0,0:03:45.34,0:03:47.77,Default,,0000,0000,0000,,when I add it to itself I'm\Nequal to negative 2?
Dialogue: 0,0:03:47.77,0:03:49.00,Default,,0000,0000,0000,,Well, we already answered\Nthat question.
Dialogue: 0,0:03:49.00,0:03:50.04,Default,,0000,0000,0000,,It's negative 2.
Dialogue: 0,0:03:50.04,0:03:55.31,Default,,0000,0000,0000,,So we get x minus 2 times\Nx minus 2 is equal to 9.
Dialogue: 0,0:03:55.31,0:03:59.35,Default,,0000,0000,0000,,Or we could have skipped this\Nstep and written x minus 2
Dialogue: 0,0:03:59.35,0:04:02.99,Default,,0000,0000,0000,,squared is equal to 9.
Dialogue: 0,0:04:02.99,0:04:07.28,Default,,0000,0000,0000,,And then you take the square\Nroot of both sides, you get x
Dialogue: 0,0:04:07.28,0:04:10.84,Default,,0000,0000,0000,,minus 2 is equal to\Nplus or minus 3.
Dialogue: 0,0:04:10.84,0:04:16.87,Default,,0000,0000,0000,,Add 2 to both sides, you get x\Nis equal to 2 plus or minus 3.
Dialogue: 0,0:04:16.87,0:04:22.44,Default,,0000,0000,0000,,That tells us that x could be\Nequal to 2 plus 3, which is 5.
Dialogue: 0,0:04:22.44,0:04:28.96,Default,,0000,0000,0000,,Or x could be equal to 2 minus\N3, which is negative 1.
Dialogue: 0,0:04:28.96,0:04:30.65,Default,,0000,0000,0000,,And we are done.
Dialogue: 0,0:04:30.65,0:04:31.84,Default,,0000,0000,0000,,Now I want to be very clear.
Dialogue: 0,0:04:31.84,0:04:34.30,Default,,0000,0000,0000,,You could have done this without\Ncompleting the square.
Dialogue: 0,0:04:34.30,0:04:37.64,Default,,0000,0000,0000,,We could've started off\Nwith x squared minus
Dialogue: 0,0:04:37.64,0:04:39.85,Default,,0000,0000,0000,,4x is equal to 5.
Dialogue: 0,0:04:39.85,0:04:42.97,Default,,0000,0000,0000,,We could have subtracted 5 from\Nboth sides and gotten x
Dialogue: 0,0:04:42.97,0:04:47.16,Default,,0000,0000,0000,,squared minus 4x minus\N5 is equal to 0.
Dialogue: 0,0:04:47.16,0:04:51.94,Default,,0000,0000,0000,,And you could say, hey, if I\Nhave a negative 5 times a
Dialogue: 0,0:04:51.94,0:04:56.19,Default,,0000,0000,0000,,positive 1, then their product\Nis negative 5 and their sum is
Dialogue: 0,0:04:56.19,0:04:57.00,Default,,0000,0000,0000,,negative 4.
Dialogue: 0,0:04:57.00,0:05:00.80,Default,,0000,0000,0000,,So I could say this is x\Nminus 5 times x plus
Dialogue: 0,0:05:00.80,0:05:02.48,Default,,0000,0000,0000,,1 is equal to 0.
Dialogue: 0,0:05:02.48,0:05:06.81,Default,,0000,0000,0000,,And then we would say that x is\Nequal to 5 or x is equal to
Dialogue: 0,0:05:06.81,0:05:07.70,Default,,0000,0000,0000,,negative 1.
Dialogue: 0,0:05:07.70,0:05:10.35,Default,,0000,0000,0000,,And in this case, this actually\Nprobably would have
Dialogue: 0,0:05:10.35,0:05:13.45,Default,,0000,0000,0000,,been a faster way to\Ndo the problem.
Dialogue: 0,0:05:13.45,0:05:16.14,Default,,0000,0000,0000,,But the neat thing about the\Ncompleting the square is it
Dialogue: 0,0:05:16.14,0:05:17.77,Default,,0000,0000,0000,,will always work.
Dialogue: 0,0:05:17.77,0:05:21.58,Default,,0000,0000,0000,,It'll always work no matter what\Nthe coefficients are or
Dialogue: 0,0:05:21.58,0:05:23.38,Default,,0000,0000,0000,,no matter how crazy\Nthe problem is.
Dialogue: 0,0:05:23.38,0:05:25.40,Default,,0000,0000,0000,,And let me prove it to you.
Dialogue: 0,0:05:25.40,0:05:28.44,Default,,0000,0000,0000,,Let's do one that traditionally\Nwould have been
Dialogue: 0,0:05:28.44,0:05:31.14,Default,,0000,0000,0000,,a pretty painful problem if\Nwe just tried to do it by
Dialogue: 0,0:05:31.14,0:05:36.20,Default,,0000,0000,0000,,factoring, especially if we\Ndid it using grouping or
Dialogue: 0,0:05:36.20,0:05:37.02,Default,,0000,0000,0000,,something like that.
Dialogue: 0,0:05:37.02,0:05:45.07,Default,,0000,0000,0000,,Let's say we had 10x squared\Nminus 30x minus
Dialogue: 0,0:05:45.07,0:05:47.53,Default,,0000,0000,0000,,8 is equal to 0.
Dialogue: 0,0:05:47.53,0:05:50.06,Default,,0000,0000,0000,,Now, right from the get-go, you\Ncould say, hey look, we
Dialogue: 0,0:05:50.06,0:05:53.28,Default,,0000,0000,0000,,could maybe divide\Nboth sides by 2.
Dialogue: 0,0:05:53.28,0:05:54.80,Default,,0000,0000,0000,,That does simplify\Na little bit.
Dialogue: 0,0:05:54.80,0:05:56.45,Default,,0000,0000,0000,,Let's divide both sides by 2.
Dialogue: 0,0:05:56.45,0:06:02.15,Default,,0000,0000,0000,,So if you divide everything\Nby 2, what do you get?
Dialogue: 0,0:06:02.15,0:06:11.99,Default,,0000,0000,0000,,We get 5x squared minus 15x\Nminus 4 is equal to 0.
Dialogue: 0,0:06:11.99,0:06:14.54,Default,,0000,0000,0000,,But once again, now we have this\Ncrazy 5 in front of this
Dialogue: 0,0:06:14.54,0:06:16.81,Default,,0000,0000,0000,,coefficent and we would have to\Nsolve it by grouping which
Dialogue: 0,0:06:16.81,0:06:20.41,Default,,0000,0000,0000,,is a reasonably painful\Nprocess.
Dialogue: 0,0:06:20.41,0:06:23.41,Default,,0000,0000,0000,,But we can now go straight to\Ncompleting the square, and to
Dialogue: 0,0:06:23.41,0:06:27.50,Default,,0000,0000,0000,,do that I'm now going to divide\Nby 5 to get a 1 leading
Dialogue: 0,0:06:27.50,0:06:28.87,Default,,0000,0000,0000,,coefficient here.
Dialogue: 0,0:06:28.87,0:06:31.66,Default,,0000,0000,0000,,And you're going to see why this\Nis different than what
Dialogue: 0,0:06:31.66,0:06:33.01,Default,,0000,0000,0000,,we've traditionally done.
Dialogue: 0,0:06:33.01,0:06:35.73,Default,,0000,0000,0000,,So if I divide this whole thing\Nby 5, I could have just
Dialogue: 0,0:06:35.73,0:06:38.05,Default,,0000,0000,0000,,divided by 10 from the get-go\Nbut I wanted to go to this the
Dialogue: 0,0:06:38.05,0:06:40.03,Default,,0000,0000,0000,,step first just to show\Nyou that this really
Dialogue: 0,0:06:40.03,0:06:41.80,Default,,0000,0000,0000,,didn't give us much.
Dialogue: 0,0:06:41.80,0:06:43.66,Default,,0000,0000,0000,,Let's divide everything by 5.
Dialogue: 0,0:06:43.66,0:06:52.69,Default,,0000,0000,0000,,So if you divide everything by\N5, you get x squared minus 3x
Dialogue: 0,0:06:52.69,0:06:58.72,Default,,0000,0000,0000,,minus 4/5 is equal to 0.
Dialogue: 0,0:06:58.72,0:07:02.02,Default,,0000,0000,0000,,So, you might say, hey, why did\Nwe ever do that factoring
Dialogue: 0,0:07:02.02,0:07:02.63,Default,,0000,0000,0000,,by grouping?
Dialogue: 0,0:07:02.63,0:07:06.14,Default,,0000,0000,0000,,If we can just always divide by\Nthis leading coefficient,
Dialogue: 0,0:07:06.14,0:07:07.22,Default,,0000,0000,0000,,we can get rid of that.
Dialogue: 0,0:07:07.22,0:07:09.84,Default,,0000,0000,0000,,We can always turn this into a 1\Nor a negative 1 if we divide
Dialogue: 0,0:07:09.84,0:07:10.91,Default,,0000,0000,0000,,by the right number.
Dialogue: 0,0:07:10.91,0:07:14.41,Default,,0000,0000,0000,,But notice, by doing that we\Ngot this crazy 4/5 here.
Dialogue: 0,0:07:14.41,0:07:17.63,Default,,0000,0000,0000,,So this is super hard to do\Njust using factoring.
Dialogue: 0,0:07:17.63,0:07:19.50,Default,,0000,0000,0000,,You'd have to say, what two\Nnumbers when I take the
Dialogue: 0,0:07:19.50,0:07:22.10,Default,,0000,0000,0000,,product is equal to\Nnegative 4/5?
Dialogue: 0,0:07:22.10,0:07:25.21,Default,,0000,0000,0000,,It's a fraction and when I take\Ntheir sum, is equal to
Dialogue: 0,0:07:25.21,0:07:26.14,Default,,0000,0000,0000,,negative 3?
Dialogue: 0,0:07:26.14,0:07:29.31,Default,,0000,0000,0000,,This is a hard problem\Nwith factoring.
Dialogue: 0,0:07:29.31,0:07:36.86,Default,,0000,0000,0000,,This is hard using factoring.
Dialogue: 0,0:07:36.86,0:07:42.08,Default,,0000,0000,0000,,So, the best thing to do is to\Nuse completing the square.
Dialogue: 0,0:07:42.08,0:07:44.72,Default,,0000,0000,0000,,So let's think a little bit\Nabout how we can turn this
Dialogue: 0,0:07:44.72,0:07:45.95,Default,,0000,0000,0000,,into a perfect square.
Dialogue: 0,0:07:45.95,0:07:48.08,Default,,0000,0000,0000,,What I like to do-- and you'll\Nsee this done some ways and
Dialogue: 0,0:07:48.08,0:07:50.04,Default,,0000,0000,0000,,I'll show you both ways because\Nyou'll see teachers do
Dialogue: 0,0:07:50.04,0:07:53.88,Default,,0000,0000,0000,,it both ways-- I like to get\Nthe 4/5 on the other side.
Dialogue: 0,0:07:53.88,0:07:56.90,Default,,0000,0000,0000,,So let's add 4/5 to both\Nsides of this equation.
Dialogue: 0,0:07:56.90,0:07:59.98,Default,,0000,0000,0000,,You don't have to do it this\Nway, but I like to get the 4/5
Dialogue: 0,0:07:59.98,0:08:01.16,Default,,0000,0000,0000,,out of the way.
Dialogue: 0,0:08:01.16,0:08:04.01,Default,,0000,0000,0000,,And then what do we get\Nif we add 4/5 to both
Dialogue: 0,0:08:04.01,0:08:05.25,Default,,0000,0000,0000,,sides of this equation?
Dialogue: 0,0:08:05.25,0:08:08.35,Default,,0000,0000,0000,,The left-hand hand side of the\Nequation just becomes x
Dialogue: 0,0:08:08.35,0:08:11.80,Default,,0000,0000,0000,,squared minus 3x,\Nno 4/5 there.
Dialogue: 0,0:08:11.80,0:08:13.66,Default,,0000,0000,0000,,I'm going to leave a little\Nbit of space.
Dialogue: 0,0:08:13.66,0:08:17.79,Default,,0000,0000,0000,,And that's going to\Nbe equal to 4/5.
Dialogue: 0,0:08:17.79,0:08:19.99,Default,,0000,0000,0000,,Now, just like the last problem,\Nwe want to turn this
Dialogue: 0,0:08:19.99,0:08:23.35,Default,,0000,0000,0000,,left-hand side into the perfect\Nsquare of a binomial.
Dialogue: 0,0:08:23.35,0:08:24.74,Default,,0000,0000,0000,,How do we do that?
Dialogue: 0,0:08:24.74,0:08:28.36,Default,,0000,0000,0000,,Well, we say, well, what number\Ntimes 2 is equal to
Dialogue: 0,0:08:28.36,0:08:30.11,Default,,0000,0000,0000,,negative 3?
Dialogue: 0,0:08:30.11,0:08:32.31,Default,,0000,0000,0000,,So some number times\N2 is negative 3.
Dialogue: 0,0:08:32.31,0:08:35.33,Default,,0000,0000,0000,,Or we essentially just take\Nnegative 3 and divide it by 2,
Dialogue: 0,0:08:35.33,0:08:37.37,Default,,0000,0000,0000,,which is negative 3/2.
Dialogue: 0,0:08:37.37,0:08:39.55,Default,,0000,0000,0000,,And then we square\Nnegative 3/2.
Dialogue: 0,0:08:39.55,0:08:44.84,Default,,0000,0000,0000,,So in the example, we'll\Nsay a is negative 3/2.
Dialogue: 0,0:08:44.84,0:08:48.38,Default,,0000,0000,0000,,And if we square negative\N3/2, what do we get?
Dialogue: 0,0:08:48.38,0:08:54.10,Default,,0000,0000,0000,,We get positive 9/4.
Dialogue: 0,0:08:54.10,0:08:56.81,Default,,0000,0000,0000,,I just took half of this\Ncoefficient, squared it, got
Dialogue: 0,0:08:56.81,0:08:58.01,Default,,0000,0000,0000,,positive 9/4.
Dialogue: 0,0:08:58.01,0:09:00.72,Default,,0000,0000,0000,,The whole purpose of doing that\Nis to turn this left-hand
Dialogue: 0,0:09:00.72,0:09:02.92,Default,,0000,0000,0000,,side into a perfect square.
Dialogue: 0,0:09:02.92,0:09:05.53,Default,,0000,0000,0000,,Now, anything you do to one side\Nof the equation, you've
Dialogue: 0,0:09:05.53,0:09:06.60,Default,,0000,0000,0000,,got to do to the other side.
Dialogue: 0,0:09:06.60,0:09:11.03,Default,,0000,0000,0000,,So we added a 9/4 here, let's\Nadd a 9/4 over there.
Dialogue: 0,0:09:11.03,0:09:13.85,Default,,0000,0000,0000,,And what does our\Nequation become?
Dialogue: 0,0:09:13.85,0:09:22.53,Default,,0000,0000,0000,,We get x squared minus 3x plus\N9/4 is equal to-- let's see if
Dialogue: 0,0:09:22.53,0:09:24.46,Default,,0000,0000,0000,,we can get a common\Ndenominator.
Dialogue: 0,0:09:24.46,0:09:29.12,Default,,0000,0000,0000,,So, 4/5 is the same\Nthing as 16/20.
Dialogue: 0,0:09:29.12,0:09:31.88,Default,,0000,0000,0000,,Just multiply the numerator\Nand denominator by 4.
Dialogue: 0,0:09:31.88,0:09:33.82,Default,,0000,0000,0000,,Plus over 20.
Dialogue: 0,0:09:33.82,0:09:36.96,Default,,0000,0000,0000,,9/4 is the same thing\Nif you multiply the
Dialogue: 0,0:09:36.96,0:09:42.15,Default,,0000,0000,0000,,numerator by 5 as 45/20.
Dialogue: 0,0:09:42.15,0:09:44.97,Default,,0000,0000,0000,,And so what is 16 plus 45?
Dialogue: 0,0:09:44.97,0:09:47.02,Default,,0000,0000,0000,,You see, this is kind of getting\Nkind of hairy, but
Dialogue: 0,0:09:47.02,0:09:48.93,Default,,0000,0000,0000,,that's the fun, I guess, of
Dialogue: 0,0:09:48.93,0:09:50.38,Default,,0000,0000,0000,,completing the square sometimes.
Dialogue: 0,0:09:50.38,0:09:53.42,Default,,0000,0000,0000,,16 plus 45.
Dialogue: 0,0:09:53.42,0:09:55.78,Default,,0000,0000,0000,,See that's 55, 61.
Dialogue: 0,0:09:55.78,0:09:59.75,Default,,0000,0000,0000,,So this is equal to 61/20.
Dialogue: 0,0:09:59.75,0:10:02.68,Default,,0000,0000,0000,,So let me just rewrite it.
Dialogue: 0,0:10:02.68,0:10:09.48,Default,,0000,0000,0000,,x squared minus 3x plus\N9/4 is equal to 61/20.
Dialogue: 0,0:10:09.48,0:10:11.03,Default,,0000,0000,0000,,Crazy number.
Dialogue: 0,0:10:11.03,0:10:13.63,Default,,0000,0000,0000,,Now this, at least on\Nthe left hand side,
Dialogue: 0,0:10:13.63,0:10:15.97,Default,,0000,0000,0000,,is a perfect square.
Dialogue: 0,0:10:15.97,0:10:21.61,Default,,0000,0000,0000,,This is the same thing as\Nx minus 3/2 squared.
Dialogue: 0,0:10:21.61,0:10:24.20,Default,,0000,0000,0000,,And it was by design.
Dialogue: 0,0:10:24.20,0:10:27.59,Default,,0000,0000,0000,,Negative 3/2 times negative\N3/2 is positive 9/4.
Dialogue: 0,0:10:27.59,0:10:32.79,Default,,0000,0000,0000,,Negative 3/2 plus negative 3/2\Nis equal to negative 3.
Dialogue: 0,0:10:32.79,0:10:37.96,Default,,0000,0000,0000,,So this squared is\Nequal to 61/20.
Dialogue: 0,0:10:37.96,0:10:43.09,Default,,0000,0000,0000,,We can take the square root of\Nboth sides and we get x minus
Dialogue: 0,0:10:43.09,0:10:47.82,Default,,0000,0000,0000,,3/2 is equal to the positive\Nor the negative
Dialogue: 0,0:10:47.82,0:10:53.32,Default,,0000,0000,0000,,square root of 61/20.
Dialogue: 0,0:10:53.32,0:10:57.64,Default,,0000,0000,0000,,And now, we can add 3/2 to both\Nsides of this equation
Dialogue: 0,0:10:57.64,0:11:03.60,Default,,0000,0000,0000,,and you get x is equal to\Npositive 3/2 plus or minus the
Dialogue: 0,0:11:03.60,0:11:07.30,Default,,0000,0000,0000,,square root of 61/20.
Dialogue: 0,0:11:07.30,0:11:09.29,Default,,0000,0000,0000,,And this is a crazy number and\Nit's hopefully obvious you
Dialogue: 0,0:11:09.29,0:11:11.43,Default,,0000,0000,0000,,would not have been able to-- at\Nleast I would not have been
Dialogue: 0,0:11:11.43,0:11:15.25,Default,,0000,0000,0000,,able to-- get to this number\Njust by factoring.
Dialogue: 0,0:11:15.25,0:11:17.26,Default,,0000,0000,0000,,And if you want their actual\Nvalues, you can get your
Dialogue: 0,0:11:17.26,0:11:18.51,Default,,0000,0000,0000,,calculator out.
Dialogue: 0,0:11:18.51,0:11:20.62,Default,,0000,0000,0000,,
Dialogue: 0,0:11:20.62,0:11:22.51,Default,,0000,0000,0000,,And then let me clear\Nall of this.
Dialogue: 0,0:11:22.51,0:11:25.95,Default,,0000,0000,0000,,
Dialogue: 0,0:11:25.95,0:11:28.76,Default,,0000,0000,0000,,And 3/2-- let's do the plus\Nversion first. So we want to
Dialogue: 0,0:11:28.76,0:11:33.71,Default,,0000,0000,0000,,do 3 divided by 2 plus the\Nsecond square root.
Dialogue: 0,0:11:33.71,0:11:35.05,Default,,0000,0000,0000,,We want to pick that little\Nyellow square root.
Dialogue: 0,0:11:35.05,0:11:46.48,Default,,0000,0000,0000,,So the square root of 61 divided\Nby 20, which is 3.24.
Dialogue: 0,0:11:46.48,0:11:52.76,Default,,0000,0000,0000,,This crazy 3.2464, I'll\Njust write 3.246.
Dialogue: 0,0:11:52.76,0:12:02.23,Default,,0000,0000,0000,,So this is approximately equal\Nto 3.246, and that was just
Dialogue: 0,0:12:02.23,0:12:03.11,Default,,0000,0000,0000,,the positive version.
Dialogue: 0,0:12:03.11,0:12:06.71,Default,,0000,0000,0000,,Let's do the subtraction\Nversion.
Dialogue: 0,0:12:06.71,0:12:09.18,Default,,0000,0000,0000,,So we can actually put our\Nentry-- if you do second and
Dialogue: 0,0:12:09.18,0:12:11.54,Default,,0000,0000,0000,,then entry, that we want that\Nlittle yellow entry, that's
Dialogue: 0,0:12:11.54,0:12:12.46,Default,,0000,0000,0000,,why I pressed the\Nsecond button.
Dialogue: 0,0:12:12.46,0:12:16.13,Default,,0000,0000,0000,,So I press enter, it puts in\Nwhat we just put, we can just
Dialogue: 0,0:12:16.13,0:12:23.40,Default,,0000,0000,0000,,change the positive or the\Naddition to a subtraction and
Dialogue: 0,0:12:23.40,0:12:27.97,Default,,0000,0000,0000,,you get negative 0.246.
Dialogue: 0,0:12:27.97,0:12:33.80,Default,,0000,0000,0000,,So you get negative 0.246.
Dialogue: 0,0:12:33.80,0:12:38.20,Default,,0000,0000,0000,,And you can actually verify\Nthat these satisfy our
Dialogue: 0,0:12:38.20,0:12:39.36,Default,,0000,0000,0000,,original equation.
Dialogue: 0,0:12:39.36,0:12:42.05,Default,,0000,0000,0000,,Our original equation\Nwas up here.
Dialogue: 0,0:12:42.05,0:12:43.84,Default,,0000,0000,0000,,Let me just verify\Nfor one of them.
Dialogue: 0,0:12:43.84,0:12:47.40,Default,,0000,0000,0000,,
Dialogue: 0,0:12:47.40,0:12:50.13,Default,,0000,0000,0000,,So the second answer on your\Ngraphing calculator is the
Dialogue: 0,0:12:50.13,0:12:51.76,Default,,0000,0000,0000,,last answer you use.
Dialogue: 0,0:12:51.76,0:12:54.16,Default,,0000,0000,0000,,So if you use a variable answer,\Nthat's this number
Dialogue: 0,0:12:54.16,0:12:55.16,Default,,0000,0000,0000,,right here.
Dialogue: 0,0:12:55.16,0:13:00.09,Default,,0000,0000,0000,,So if I have my answer squared--\NI'm using answer
Dialogue: 0,0:13:00.09,0:13:02.38,Default,,0000,0000,0000,,represents negative 0.24.
Dialogue: 0,0:13:02.38,0:13:11.98,Default,,0000,0000,0000,,Answer squared minus 3 times\Nanswer minus 4/5-- 4 divided
Dialogue: 0,0:13:11.98,0:13:16.03,Default,,0000,0000,0000,,by 5-- it equals--.
Dialogue: 0,0:13:16.03,0:13:18.49,Default,,0000,0000,0000,,And this just a little\Nbit of explanation.
Dialogue: 0,0:13:18.49,0:13:21.86,Default,,0000,0000,0000,,This doesn't store the entire\Nnumber, it goes up to some
Dialogue: 0,0:13:21.86,0:13:22.88,Default,,0000,0000,0000,,level of precision.
Dialogue: 0,0:13:22.88,0:13:24.91,Default,,0000,0000,0000,,It stores some number\Nof digits.
Dialogue: 0,0:13:24.91,0:13:28.93,Default,,0000,0000,0000,,So when it calculated it using\Nthis stored number right here,
Dialogue: 0,0:13:28.93,0:13:32.24,Default,,0000,0000,0000,,it got 1 times 10 to\Nthe negative 14.
Dialogue: 0,0:13:32.24,0:13:34.98,Default,,0000,0000,0000,,So that is 0.0000.
Dialogue: 0,0:13:34.98,0:13:37.10,Default,,0000,0000,0000,,So that's 13 zeroes\Nand then a 1.
Dialogue: 0,0:13:37.10,0:13:38.87,Default,,0000,0000,0000,,A decimal, then 13\Nzeroes and a 1.
Dialogue: 0,0:13:38.87,0:13:41.06,Default,,0000,0000,0000,,So this is pretty much 0.
Dialogue: 0,0:13:41.06,0:13:43.55,Default,,0000,0000,0000,,Or actually, if you got the\Nexact answer right here, if
Dialogue: 0,0:13:43.55,0:13:46.48,Default,,0000,0000,0000,,you went through an infinite\Nlevel of precision here, or
Dialogue: 0,0:13:46.48,0:13:49.05,Default,,0000,0000,0000,,maybe if you kept it in this\Nradical form, you would get
Dialogue: 0,0:13:49.05,0:13:52.39,Default,,0000,0000,0000,,that it is indeed equal to 0.
Dialogue: 0,0:13:52.39,0:13:55.30,Default,,0000,0000,0000,,So hopefully you found that\Nhelpful, this whole notion of
Dialogue: 0,0:13:55.30,0:13:56.16,Default,,0000,0000,0000,,completing the square.
Dialogue: 0,0:13:56.16,0:13:58.67,Default,,0000,0000,0000,,Now we're going to extend it\Nto the actual quadratic
Dialogue: 0,0:13:58.67,0:14:01.51,Default,,0000,0000,0000,,formula that we can use, we\Ncan essentially just plug
Dialogue: 0,0:14:01.51,0:14:03.61,Default,,0000,0000,0000,,things into to solve any\Nquadratic equation.
Dialogue: 0,0:14:03.61,0:14:05.27,Default,,0000,0000,0000,,