04 - Completing the Square to Solve Quadratic Equations - Part 1 - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . The name of this election | |
| 00:02 | is called completing the square . This is part one | |
| 00:05 | . This section is actually so incredibly important that we're | |
| 00:08 | going to have not one , not two , not | |
| 00:10 | three not four but five parts to this lesson . | |
| 00:13 | In this lesson we're gonna learn about what completing the | |
| 00:15 | square is . We're gonna go through a couple of | |
| 00:17 | examples to make sure you absolutely understand how to do | |
| 00:20 | it . But then in the next several lessons , | |
| 00:22 | the next four we're just going to increase the problem | |
| 00:24 | complexity , making them harder and harder and harder to | |
| 00:27 | actually use the technique . But we're doing exactly the | |
| 00:31 | same steps that we're actually going to learn in this | |
| 00:33 | lesson . So we know that quadratic equations are the | |
| 00:36 | most one of the most common types of equations we | |
| 00:38 | solve in real science and math . We know that | |
| 00:40 | we really want to solve those types of equations . | |
| 00:43 | We've learned how to solve some quadratic equations in the | |
| 00:45 | past we've learned specifically how to factor them and solve | |
| 00:48 | them by factoring and we've learned the very special ones | |
| 00:51 | , we call them perfect square quadratic . In the | |
| 00:53 | last lesson where we could just basically take the square | |
| 00:56 | root of both sides . What we're gonna do here | |
| 00:59 | is learn how to solve any quadratic you get that | |
| 01:02 | . I give you I can give you literally any | |
| 01:04 | quadratic I can think of and you will be able | |
| 01:06 | to complete the square in order to solve it . | |
| 01:09 | And this concept of completing the square is going to | |
| 01:12 | tie directly into the next topic we're going to talk | |
| 01:15 | about , which is the very famous quadratic formula . | |
| 01:18 | I'm just going to give you a little bit of | |
| 01:19 | a punchline . The quadratic formula comes from this concept | |
| 01:23 | to completing the square . In other words , I'm | |
| 01:25 | going to derive the quadratic formula and show you where | |
| 01:28 | the quadratic formula comes from . In the next lesson | |
| 01:30 | in the next topic , from using the concept of | |
| 01:33 | completing the square . So that is how important it | |
| 01:35 | is . It actually is used to derive , complete | |
| 01:38 | the quadratic formula , which is one of the most | |
| 01:40 | famous formulas in all of algebra . So let's crawl | |
| 01:43 | before we walk . We want to talk about what | |
| 01:45 | we're doing and what we know from the past and | |
| 01:48 | what this is and how it ties into what we're | |
| 01:50 | going in the future . So we know that we | |
| 01:52 | can solve by factoring so some of these things we | |
| 01:54 | can solve by factoring right ? And you know , | |
| 02:00 | we've done this many , many times so much so | |
| 02:03 | that you might not even realize that it was a | |
| 02:04 | solution technique . But for instance , if I gave | |
| 02:07 | you this equation , which we've done this before , | |
| 02:09 | all this stuff is stuff we've done X squared plus | |
| 02:11 | two X plus one . Now this is a quadratic | |
| 02:15 | equation , you know that because it's an X squared | |
| 02:17 | and we're setting this equal to zero . So what | |
| 02:19 | we're doing is if you were to plot this quadratic | |
| 02:21 | equation , it would have some crossing points on the | |
| 02:24 | X axis . It turns out that there might not | |
| 02:26 | be crossing points there , in which case you have | |
| 02:29 | imaginary solutions , but if we did have two crossing | |
| 02:31 | points , those would be the real roots of this | |
| 02:33 | equation . And if it was hovering above the axis | |
| 02:36 | or below , then we would have the imaginary solutions | |
| 02:38 | because there would be no crossing points at all . | |
| 02:40 | Right . Um But the way that we learned how | |
| 02:43 | to solve this in the past was to first try | |
| 02:45 | to factor it . So we draw the two binomial | |
| 02:47 | and set it equal to zero . And we say | |
| 02:49 | , well we have an X here in an X | |
| 02:50 | . Here to multiply give me X squared . Only | |
| 02:53 | way I can get a one is one times one | |
| 02:55 | . So the only way this is going to really | |
| 02:57 | work out is with a plus sign in a plus | |
| 02:58 | sign . Right ? So we've learned how to do | |
| 03:01 | this before because you can multiply it out . This | |
| 03:03 | is going to give you two X . This is | |
| 03:05 | going to give you , I'm sorry , one X | |
| 03:06 | . This is going to give you one X . | |
| 03:07 | You add them together and get to X . The | |
| 03:09 | last term's multiply to give you one the first times | |
| 03:11 | multiplayer give you give you the X square . So | |
| 03:14 | this is the fact terrible form . And you learn | |
| 03:16 | that to solve this guy , you just say X | |
| 03:18 | plus one . You can test that this term equal | |
| 03:20 | to zero and then of course it's identical . But | |
| 03:23 | you can say that this other term here is equal | |
| 03:25 | to zero . So then you can say x is | |
| 03:26 | negative one , X is uh negative one like this | |
| 03:31 | . All right , now , what does that physically | |
| 03:32 | mean ? It's a double root , right ? We | |
| 03:34 | have two routes , but they're exactly the same thing | |
| 03:37 | and we learned about that in the past and we | |
| 03:39 | just go over and kind of drawn aside here , | |
| 03:42 | if we were to draw this thing , we're saying | |
| 03:44 | that this thing has roots at negative one , negative | |
| 03:46 | one is somewhere here . And that means that the | |
| 03:49 | graph of this thing . If you were to draw | |
| 03:51 | , it would go down and then touch and then | |
| 03:53 | go up something like this . So I have a | |
| 03:56 | double root right at this point . But the point | |
| 03:58 | is is that we learn how to solve this by | |
| 04:00 | factoring . So the very first thing you do , | |
| 04:02 | when you see a quadratic is you try to factor | |
| 04:04 | it . But as you know , you can't factor | |
| 04:06 | some of these things so you can't do this unless | |
| 04:08 | you know how to factor it . So let's give | |
| 04:10 | an example of when you don't know what to do | |
| 04:12 | , because you don't know how to factor it . | |
| 04:13 | For instance , if you can't factor something , can't | |
| 04:17 | factor the following equation . What if I give you | |
| 04:23 | X squared plus two X plus nine ? So this | |
| 04:26 | is exactly the same form as this one . Uh | |
| 04:29 | And of course we're setting it equal to zero and | |
| 04:31 | we would like to solve it . The first thing | |
| 04:33 | you do is you try to factor it . So | |
| 04:34 | you put your princes here , you set it equal | |
| 04:37 | to zero . The X squared is exactly the same | |
| 04:39 | . And then you have to ask yourself what times | |
| 04:41 | what is nine ? So the only choices I really | |
| 04:43 | have is three times three . If I put a | |
| 04:45 | three here in a three here then I'm gonna have | |
| 04:47 | a three X . And three X . That's gonna | |
| 04:49 | give me a six X . So that's not right | |
| 04:51 | . And then if I change this around instead of | |
| 04:53 | three times three and make it one times nine which | |
| 04:55 | also give me nine , it'll be 91 X . | |
| 04:58 | And nine X . Which which there's no way you | |
| 05:00 | can get it to even if you change the signs | |
| 05:02 | it's not gonna work . So you cannot factor this | |
| 05:05 | . Now this does not mean that this has no | |
| 05:07 | solution . It just means that you can't solve it | |
| 05:10 | by factoring because there's no way to factor with whole | |
| 05:13 | numbers like this with whole integral numbers . There's no | |
| 05:16 | way to get it all to work out so that | |
| 05:17 | you can factor and then set everything equal to zero | |
| 05:20 | . So do you give up , know what we're | |
| 05:22 | going to learn is a technique where I can solve | |
| 05:25 | this equation . No problem . But I just can't | |
| 05:27 | do it by factoring because I don't know how to | |
| 05:29 | factor it . So keep this in the back of | |
| 05:31 | your mind . Let's go over here and talk about | |
| 05:34 | the following thing . Let me draw a little divider | |
| 05:36 | bar here and let me remind you in the last | |
| 05:40 | few lessons we've learned how to solve these things . | |
| 05:42 | We can solve what we called Perfect square quadratic . | |
| 05:55 | These were called perfect square quadratic . And what did | |
| 05:57 | they look like ? Let's give a couple of quick | |
| 05:59 | examples . The perfect square quadratic with two x minus | |
| 06:03 | three , quantity squared is equal to seven . And | |
| 06:06 | another example would be something like X plus five quantity | |
| 06:11 | squared is equal to negative four . We know how | |
| 06:14 | to solve both of these things . These are called | |
| 06:16 | perfect square quadratic . We just did it in the | |
| 06:18 | last lesson . How do we solve them ? Well | |
| 06:20 | we have the variable tied up on one side with | |
| 06:23 | a square . So we take a square root of | |
| 06:25 | both sides that reveals the variable and allows us to | |
| 06:28 | manipulate and solve , we take the square into both | |
| 06:30 | sides here . But the problem is we have a | |
| 06:32 | negative four . So when we take the square root | |
| 06:34 | of that side we're gonna get an imaginary number , | |
| 06:36 | we'll get to I so when we solve this one | |
| 06:39 | by taking the square root , we're gonna get imaginary | |
| 06:41 | results . When we take the square root of this | |
| 06:43 | side of this guy and solve , we're gonna get | |
| 06:45 | real answers . But the point is is we know | |
| 06:47 | how to solve both of them . But the only | |
| 06:49 | reason that we can solve it is because this thing | |
| 06:51 | is in a form so that I can just take | |
| 06:53 | the square root and neatly cancel everything related to this | |
| 06:57 | variable . I can't take the square to both sides | |
| 06:59 | of this because the variable is in two locations . | |
| 07:02 | But I can do it here because it's nice and | |
| 07:04 | wrapped up like this . So what we're gonna learn | |
| 07:07 | about this lesson is called completing the square . And | |
| 07:09 | it just is a technique that lets us start from | |
| 07:13 | an equation like this , which we cannot factor and | |
| 07:15 | we cannot take the square root of both sides , | |
| 07:17 | we cannot solve and we're going to change it so | |
| 07:20 | that it looks like a perfect square quadratic , which | |
| 07:22 | we do know how to solve . I'm gonna say | |
| 07:24 | that two more times because it's so important when I | |
| 07:27 | first learned completing the square , I didn't understand why | |
| 07:30 | we were learning it . The reason you're learning completing | |
| 07:33 | the square is because up until now in algebra you | |
| 07:35 | only know how to solve very special quadratic equations . | |
| 07:39 | You can only solve them if they're factory bill like | |
| 07:42 | this and you can only solve them if they're very | |
| 07:45 | , very easily able to take the square to both | |
| 07:47 | sides like this . But you do not know up | |
| 07:49 | until this point in algebra had to solve this because | |
| 07:51 | you cannot factor and you cannot take the square to | |
| 07:53 | both sides to neatly reveal the variable because the variables | |
| 07:57 | in two locations with different powers . So what we're | |
| 07:59 | gonna do is learn a technique that can take any | |
| 08:02 | equation like this . It doesn't have to be this | |
| 08:03 | one . It could be any pollen on any quadratic | |
| 08:06 | and change it to look like a perfect square quadratic | |
| 08:09 | , which means I can just take the square on | |
| 08:11 | both sides . That is really important . The third | |
| 08:13 | time I will take any quadratic polynomial and converted to | |
| 08:17 | a perfect square quadratic . And that's why we learned | |
| 08:19 | that topic first because we're gonna use it here . | |
| 08:22 | So there's a little technique to it , there's a | |
| 08:25 | few steps to it . And instead of writing all | |
| 08:27 | the steps down , we're gonna go through two examples | |
| 08:29 | really carefully and I'm not gonna write a lot of | |
| 08:32 | words but I'm going to do a lot of talking | |
| 08:34 | . But it's really important for you to follow every | |
| 08:36 | step and get to the end . And even at | |
| 08:38 | the end of it , you might think , well | |
| 08:39 | how do I know it works for everything ? How | |
| 08:41 | do I know it always works ? Well that's not | |
| 08:43 | your job . The mathematicians have proven that completing the | |
| 08:46 | square will always work so you can solve any quadratic | |
| 08:50 | using completing the square , it's up to you to | |
| 08:52 | be able to follow the recipe , so to speak | |
| 08:55 | . So what we wanna do is want to solve | |
| 08:57 | this equation which is not fact arable . Let's say | |
| 08:59 | it's X squared minus six X minus three and that's | |
| 09:04 | equal to zero . Well the very first thing you | |
| 09:06 | want to try to do any time you're given one | |
| 09:08 | of these out of the blue is try to factor | |
| 09:10 | it . Always try that first . So we're gonna | |
| 09:13 | do this instead of equal to zero . So we | |
| 09:14 | have an X . Squared . So we'll put an | |
| 09:16 | X . In the next year and then we look | |
| 09:18 | and we have a three . So we can only | |
| 09:20 | do one times three . That's the only way you | |
| 09:22 | can make three . So you pretty much have to | |
| 09:23 | choose one and a three here and you try to | |
| 09:25 | pick plus minus minus plus . And there's no way | |
| 09:28 | to do it because you can have a one X | |
| 09:29 | in the middle and a three X from the outside | |
| 09:32 | . There's no way to add or subtract those to | |
| 09:34 | get negative six X . So this is not fact | |
| 09:37 | herbal , so not factor will , so before this | |
| 09:44 | you would just give up , you would have no | |
| 09:45 | idea how to solve it . But completing the square | |
| 09:47 | is going to allow us to start with this and | |
| 09:50 | end up with a perfect square quadratic that we do | |
| 09:53 | know how to solve . And it's a five step | |
| 09:57 | process but actually it's only three steps . The last | |
| 10:00 | couple of steps is just solving solving the perfect square | |
| 10:03 | quadratic that comes out of it . So actually completing | |
| 10:05 | the square is only three steps . Um but for | |
| 10:07 | some reason it trips up a lot of students . | |
| 10:09 | Um So what we're gonna do is take it slow | |
| 10:11 | and make sure you understand it . Step number one | |
| 10:14 | . What we wanna do , we have this quadratic | |
| 10:17 | equal to zero . What you want to do is | |
| 10:18 | take whatever constant number in this case we have a | |
| 10:21 | negative three . You want to take whatever is constant | |
| 10:22 | and you want to move it to the other side | |
| 10:24 | of the equal side . So you want to add | |
| 10:27 | three to both sides . You want to just get | |
| 10:32 | rid of whatever constant . So when I say add | |
| 10:33 | three to both sides , I'm not saying that you | |
| 10:35 | always add three for every equation . I'm saying you | |
| 10:37 | add whatever constant you have at the end of the | |
| 10:41 | equation , you just move it to the other side | |
| 10:43 | . If it were positive for you would be subtracting | |
| 10:46 | for from both sides . All you want to do | |
| 10:47 | is take that constant and move it to the other | |
| 10:49 | side . Using addition and subtraction . In this case | |
| 10:52 | we're gonna be adding three to both sides . So | |
| 10:54 | the result of that is the following thing , X | |
| 10:57 | squared minus six X . Is equal to three . | |
| 10:59 | All we did was add three here makes it go | |
| 11:01 | away , add three here , pops three over there | |
| 11:04 | . So you all know how to add step # | |
| 11:06 | one is really easy . The only one that gives | |
| 11:09 | people problems is step two because it sounds really complicated | |
| 11:13 | . All right . So what you need to do | |
| 11:15 | , you know what in order to do ? Step | |
| 11:17 | to , What I want to do is I want | |
| 11:19 | to rewrite this . Actually , let's see here first | |
| 11:24 | . I'm gonna just go and I'm gonna market down | |
| 11:25 | from what I have above . So what you want | |
| 11:27 | to do is you want to take this guy , | |
| 11:29 | you look at the coefficient that's in front of the | |
| 11:31 | X term . In this case it's a negative six | |
| 11:34 | . And what you wanna do is divided by two | |
| 11:36 | , so negative six , divided by two . All | |
| 11:38 | you do is you take what's here and you divide | |
| 11:39 | it by two . Notice I had a negative here | |
| 11:41 | because it's a negative in my original equation . You | |
| 11:44 | divide it by two and then you take whatever you | |
| 11:46 | have and you square it right , you'll understand as | |
| 11:50 | we get to the end of . So what you're | |
| 11:51 | gonna do is you're gonna look at what you got | |
| 11:52 | from step one divided by two and square it . | |
| 11:55 | And then you're going to add this to both sods | |
| 12:01 | . Now it turns out in this case that negative | |
| 12:05 | 6/2 is very easy . So this actually comes out | |
| 12:07 | to be what negative three , quantity square negative three | |
| 12:10 | times negative three is actually positive nine . So we're | |
| 12:13 | not actually going to add this fraction . What we | |
| 12:15 | do is we calculate it we get a nine . | |
| 12:17 | So we add nine to both sides . Right ? | |
| 12:20 | So then what we get is the the answer or | |
| 12:22 | to the solution of step two or the end of | |
| 12:25 | step two is we take what we had the previous | |
| 12:28 | step X squared minus six X . And we add | |
| 12:31 | nine to the left and add nine to the right | |
| 12:34 | . We had a three . So we add nine | |
| 12:36 | to both sides . Notice that we haven't changed the | |
| 12:38 | equation because remember you can after subtract anything you want | |
| 12:41 | to to both sides of an equation . I'm just | |
| 12:43 | choosing to add a nine because it's been proven that | |
| 12:46 | when you do that , you make this equation really | |
| 12:48 | easy to solve if you always take this guy divided | |
| 12:51 | by two and square it . And if you add | |
| 12:53 | that to both sides , the whole thing becomes easy | |
| 12:55 | and we'll see how in a second . So what | |
| 12:57 | we have is x squared minus six X plus nine | |
| 13:00 | is equal to 12 . Now how did we make | |
| 13:04 | this easier ? Because in the original problem we did | |
| 13:08 | not know how to factor this . It was intractable | |
| 13:11 | . But now I'm telling you that after you complete | |
| 13:14 | the square , this thing on the left hand side | |
| 13:16 | is always fact herbal , we always can factor it | |
| 13:19 | because of what you've done in this process . So | |
| 13:22 | let's try it . So here you go . Step | |
| 13:24 | # three factor . So what you do is you | |
| 13:28 | go ahead and open up your princess like you always | |
| 13:30 | do and it's equal to 12 because that's what's here | |
| 13:33 | and then we have X times X . And then | |
| 13:35 | here you have nine . And a way to factor | |
| 13:37 | this is to say three times three and you have | |
| 13:39 | to choose the signs correctly , it has to be | |
| 13:41 | negative times negative . Now make sure you understand that | |
| 13:44 | This is actually the factored form of this X times | |
| 13:46 | X is x squared negative three times negative three is | |
| 13:49 | positive nine . This gives you negative three X . | |
| 13:51 | This gives you negative three X . Which gives you | |
| 13:53 | negative six X . Now that's the factored form but | |
| 13:56 | it's actually a simpler factored form to just say the | |
| 13:59 | following x minus three times itself is just x minus | |
| 14:03 | three squared Is equal to 12 . Now this should | |
| 14:07 | look familiar to you because we said we know how | |
| 14:10 | to solve these things called perfect square quadratic . So | |
| 14:13 | we did a whole two lessons on it previously . | |
| 14:15 | If I give you an equation like this , we | |
| 14:17 | said it was a perfect square quadratic because I could | |
| 14:20 | just take the square root of the left and the | |
| 14:21 | right and it would basically reveal the variables and make | |
| 14:24 | it very easy to solve in one step . So | |
| 14:27 | now what we've done is we've come and started from | |
| 14:29 | an ugly expression that we don't know how to factor | |
| 14:31 | . We've done all this stuff and now we have | |
| 14:33 | a perfect square quadratic which we can always solve . | |
| 14:36 | We just take the square to both sides . So | |
| 14:38 | let's go off to the next board and I'm gonna | |
| 14:41 | write that perfect square quadratic down one more time . | |
| 14:44 | It's gonna be x minus three , quantity squared , | |
| 14:48 | it's 12 . So now we just take the square | |
| 14:49 | to both sides on the left . We take the | |
| 14:53 | square root will be x minus three left over on | |
| 14:55 | the right , it's gonna be plus or minus the | |
| 14:57 | square root of 12 . And you go over here | |
| 14:58 | and double check yourself . That's gonna be two times | |
| 15:00 | six . This is gonna be two times three , | |
| 15:02 | here's your pair right here And so what you have | |
| 15:06 | is X -3 equals plus or minus the two comes | |
| 15:11 | out and the square root of three which is left | |
| 15:14 | over remains . And now the salt for actually moved | |
| 15:16 | three over X is equal to three plus or minus | |
| 15:20 | two times the square root of three . This is | |
| 15:22 | the final answer . Three plus or minus two times | |
| 15:25 | square to three . And you can write that as | |
| 15:27 | three plus two times square to three and three minus | |
| 15:31 | two times square to three . You have two solutions | |
| 15:33 | . They're both real , which means if we were | |
| 15:35 | to graph the original quadratic that we had , we | |
| 15:38 | know that that graph is going to cross either as | |
| 15:40 | a smiley face , going dipping below the X axis | |
| 15:43 | , crossing in those two locations or going upside down | |
| 15:46 | , crossing in those two locations . But we know | |
| 15:48 | that it crosses somewhere because we figured out what those | |
| 15:51 | crossing points are . So what I want to do | |
| 15:54 | is go over it one more time and we're gonna | |
| 15:56 | do one more problem . And then over the next | |
| 15:58 | four lessons we're going to do tons more to give | |
| 16:01 | you more practice with it . But the bottom line | |
| 16:03 | is the punch line is we know how to solve | |
| 16:06 | quadratic equations by factoring if we can factor , we | |
| 16:09 | always do that , but some equations we don't know | |
| 16:11 | how to factor , so we don't know what to | |
| 16:13 | do but we also know that we can solve these | |
| 16:15 | perfect square quadratic . So this process takes any quadratic | |
| 16:19 | and turns it into a perfect square quadratic . The | |
| 16:22 | very first step is you take whatever constant term you | |
| 16:24 | have and you move it to the other side . | |
| 16:26 | I actually didn't write one other step down here as | |
| 16:29 | well , it should be kind of in between Step | |
| 16:31 | one . Step two , I'm gonna call it step | |
| 16:34 | one a And I'm going to say that the coefficient | |
| 16:38 | of the highest power here of this guy right here | |
| 16:43 | needs to be at one . So the coefficient of | |
| 16:48 | X square term needs to be positive one , only | |
| 16:56 | a one . So for instance , if this guy | |
| 16:59 | were two X squared and so on and so on | |
| 17:01 | , I would do everything the same , but I | |
| 17:02 | would have one other step in the middle where I'd | |
| 17:04 | have to divide left and right by two just to | |
| 17:07 | get this guy to be a one . So there's | |
| 17:09 | really an extra step in there . I'm sorry , | |
| 17:11 | I didn't write it down initially . But basically the | |
| 17:13 | very first step is you have to add three or | |
| 17:16 | add to get your constant to the other side . | |
| 17:18 | The next step is double check and see if the | |
| 17:20 | coefficient of this is one in this case it is | |
| 17:22 | . So we didn't have to do anything . If | |
| 17:23 | it's not one , then you have to divide both | |
| 17:25 | sides by whatever is in front to get rid of | |
| 17:27 | it . Then the next step is you take a | |
| 17:29 | look at what's in the middle in front of X | |
| 17:32 | , you divide it by two , you square it | |
| 17:34 | figure out what that is , you added to both | |
| 17:36 | sides , that's what we did here . And when | |
| 17:38 | you do that , you will always be able to | |
| 17:39 | factor this and furthermore , you will always be able | |
| 17:42 | to factor it where it'll be exactly identical twins here | |
| 17:45 | in the binomial . So you will always be able | |
| 17:47 | to write it as a perfect square quadratic . And | |
| 17:51 | then the rest of it was just solving that , | |
| 17:52 | which we've done many times before . So we want | |
| 17:55 | to go to the last board and do one more | |
| 17:58 | to make sure you understand . And then as I | |
| 18:00 | said , we're going to have a ton of other | |
| 18:01 | examples to to increase the complexity . So let's solve | |
| 18:05 | this guy . The equation we want to solve is | |
| 18:07 | X squared minus two . X minus five equals zero | |
| 18:13 | . And the first thing , of course you want | |
| 18:14 | to do is try to factor it . So let's | |
| 18:15 | go off to the side here and see if we | |
| 18:17 | can , we'll try to factor it . The X | |
| 18:20 | squared means we're gonna have X and X . The | |
| 18:22 | only way I can get five is one times five | |
| 18:25 | , So I can do pluses or I can do | |
| 18:27 | minuses or I can do any combination , but there's | |
| 18:29 | no way I can get it to because 5 -1 | |
| 18:32 | , there's no way it's gonna ever give me a | |
| 18:33 | two , so it's not factual . So we're going | |
| 18:35 | to try to solve it by completing the square . | |
| 18:37 | So the very first step in this process Is to | |
| 18:42 | take whatever constant term in this case it's a -5 | |
| 18:44 | and want to move it to the other side , | |
| 18:46 | so it's gonna be X squared minus two . X | |
| 18:49 | equals positive five . We add five to the left | |
| 18:52 | , add five to the right , and that's step | |
| 18:54 | one . Okay , Step # two . Which I | |
| 18:57 | have the steps number differently over there . We want | |
| 18:59 | to check that the coefficient of X squared is equal | |
| 19:04 | to one and it is it's already a one here | |
| 19:06 | , so I don't have to do anything . If | |
| 19:08 | this were three X squared then this would be a | |
| 19:10 | three X squared minus two X equals five . Then | |
| 19:12 | I would have to divide the left and the right | |
| 19:13 | by three . And that would get rid of the | |
| 19:15 | three in front of the X squared , making it | |
| 19:17 | a one . And then I would proceed , we're | |
| 19:18 | going to have problems like that very soon . The | |
| 19:21 | next step is we have to do and take a | |
| 19:24 | look at what's in front of this guy . So | |
| 19:26 | we take a look at what's in front of the | |
| 19:28 | X term in this case it's negative too . So | |
| 19:31 | we're going to add we're gonna take the negative two | |
| 19:34 | . We divide it by two and we square it | |
| 19:36 | . Whatever is in front of here , you divided | |
| 19:38 | by two and then you square it which is going | |
| 19:41 | to be equal to that's negative two , divided by | |
| 19:43 | two is negative one . We're gonna square that . | |
| 19:45 | What is negative one squared ? It's positive one to | |
| 19:47 | both sides . So whatever this comes out to be | |
| 19:51 | when you divide by two and square it you add | |
| 19:53 | that to both sides . So in order to do | |
| 19:56 | that , we're gonna add it here . It'll be | |
| 19:57 | x squared minus two . X plus one equals five | |
| 20:02 | plus one . You all know that ? That's six | |
| 20:04 | . So we have X squared minus two X plus | |
| 20:06 | one equals six . And the claim is that once | |
| 20:10 | you do that this is always factory and it's always | |
| 20:12 | going to be a perfect square . If you cannot | |
| 20:15 | factor this then you've done something wrong . So then | |
| 20:18 | we factor and what we're going to get here , | |
| 20:22 | we're gonna try to factor it . It should be | |
| 20:24 | equal to six . So we have now X times | |
| 20:27 | X . The only way I can get a one | |
| 20:29 | is one times one . And the only way I | |
| 20:31 | get that negative two in there is with a negative | |
| 20:33 | and a negative . Then I have a negative X | |
| 20:36 | . And a negative X adding to give me negative | |
| 20:38 | two X . These multiply to give me a positive | |
| 20:40 | one . These multiply to give me the X . | |
| 20:42 | Square . And so now I can see they're identical | |
| 20:46 | twins . That's what I should always get . So | |
| 20:47 | it's going to be X -1 quantity squared is equal | |
| 20:50 | to six . Now this is a perfect square quadratic | |
| 20:53 | . We just learned how to solve those so we're | |
| 20:55 | going to solve and so what we're going to get | |
| 20:58 | is take the square root of the left . The | |
| 21:01 | squares of the left is going to reveal X -1 | |
| 21:03 | by itself plus or minus the square root of six | |
| 21:07 | on the right . But I know that six cannot | |
| 21:09 | we cannot simplify the square root of six any further | |
| 21:11 | because all I can do is two times three and | |
| 21:13 | I can't find any pairs . So then you just | |
| 21:16 | move the one over making it positive one plus or | |
| 21:19 | minus square root of six . So that means I | |
| 21:22 | have to answer is one plus the square root of | |
| 21:24 | six and one minus the square root of six . | |
| 21:26 | I got real answers which means this original equation does | |
| 21:29 | cross the X axis some sort of way it doesn't | |
| 21:31 | hover above or below and I use completing the square | |
| 21:34 | to figure it out . So the punch line is | |
| 21:37 | something that I really want to emphasize because it's not | |
| 21:40 | something I learned when I first learned this . The | |
| 21:42 | reason we learn completing the square is because up until | |
| 21:45 | now you've only been able to factor quadratic to solve | |
| 21:48 | them or if they were very special perfect squares , | |
| 21:50 | you can solve them by taking the square to both | |
| 21:52 | sides . This process allows you to take any quadratic | |
| 21:55 | anything I give you and you can factor it in | |
| 21:58 | a way that allows you to solve it even if | |
| 22:00 | there's fractions or if there's decimals or other things going | |
| 22:04 | on , if you follow this process , you will | |
| 22:06 | be able to factor and you will be able to | |
| 22:08 | solve it and it is so powerful that completing the | |
| 22:11 | square is the way I'm gonna actually derive the quadratic | |
| 22:14 | formula , which you know , we're all going to | |
| 22:15 | use the quadratic formula going on throughout algebra to solve | |
| 22:18 | these things as well . But it comes from completing | |
| 22:20 | the square . So following on to the next few | |
| 22:22 | lessons , we're gonna increase the complexity following this procedure | |
| 22:25 | and then once you're comfortable that will move on to | |
| 22:27 | the quadratic formula to solve quadratic equations in algebra . |
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