14 - The Discriminant of a Quadratic, Part 1 (Quadratic Formula Problems) - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to algebra . The title of | |
| 00:02 | this lesson is called the Discriminative of a quadratic . | |
| 00:05 | It's part one . It's a complicated sounding title but | |
| 00:08 | I'm actually really excited to teach this lesson because I | |
| 00:10 | have a really cool computer demo . I'm going to | |
| 00:13 | show you about a few minutes into it . That's | |
| 00:15 | gonna really make what I'm about to talk about visually | |
| 00:17 | just pop out and so you can really understand it | |
| 00:20 | . So the bottom line here is we've been using | |
| 00:22 | the quadratic formula for the last several lessons to solve | |
| 00:25 | quadratic equations by now . You should be really good | |
| 00:27 | at using the quadratic formula so we can get the | |
| 00:29 | exact answers to any quadratic equation . We want to | |
| 00:33 | just by putting the A . B and C into | |
| 00:35 | the quadratic formula . We always get the two answers | |
| 00:38 | , but it turns out that we can actually learn | |
| 00:41 | a lot about the solutions of the quadratic without actually | |
| 00:46 | cranking through the entire quadratic formula . In other words | |
| 00:49 | , we may not always care about the exact solutions | |
| 00:52 | , but sometimes we may want to understand the nature | |
| 00:55 | or some characteristics of those solutions without going through the | |
| 00:58 | whole entire quadratic formula deal . So in order to | |
| 01:01 | understand that we need to kind of break apart the | |
| 01:03 | quadratic formula a little bit and talk about one part | |
| 01:06 | of the quadratic formula that we're going to call the | |
| 01:08 | discriminate . And based on how that discriminate looks , | |
| 01:11 | we can actually learn a lot about the solutions of | |
| 01:13 | the quadratic equation . Well that actually cranking through it | |
| 01:16 | all . So I want to break out what the | |
| 01:18 | quadratic formula is . I'm gonna show you what that | |
| 01:20 | discriminate is here briefly on the board and then we're | |
| 01:22 | gonna go over to the computer where I can show | |
| 01:24 | you visually what is happening when that discriminate changes with | |
| 01:28 | a graph of a quadratic function . So stick with | |
| 01:32 | me to that point and you'll definitely see an awesome | |
| 01:34 | little little demo of that . So we have generally | |
| 01:38 | any quadratic equations going to generally look like this . | |
| 01:41 | A X squared plus B . X plus C is | |
| 01:44 | equal to zero . And by now , you know | |
| 01:45 | that A . And B and C can actually be | |
| 01:48 | real numbers . That can actually be square roots , | |
| 01:51 | that can be rational , irrational numbers . They can | |
| 01:53 | also be imaginary numbers , A . B and C | |
| 01:55 | . We've done some problems where they're actually imaginary , | |
| 01:57 | but for the purpose of this lesson right now we're | |
| 02:00 | going to just say that A and B and C | |
| 02:02 | are are real numbers . They can be negative , | |
| 02:04 | they can be positive . They can be fractions . | |
| 02:06 | They can be radicals but they're definitely just gonna be | |
| 02:08 | real . All right . So we know that we | |
| 02:11 | can solve this quadratic equation using the quadratic formula . | |
| 02:14 | We're gonna get to solutions the first one We're gonna | |
| 02:17 | call it x one is gonna be negative B plus | |
| 02:20 | or minus . Actually going to take away that minus | |
| 02:22 | for now negative B plus b squared minus four times | |
| 02:26 | a times C . The radical goes around this guy | |
| 02:28 | all over to a that was solution number one . | |
| 02:33 | The only difference between the other solution is this plus | |
| 02:35 | changes to a minus . So we're gonna call the | |
| 02:37 | next solution except to and it's gonna be negative b | |
| 02:40 | minus B squared minus four times . Hmc Radical goes | |
| 02:45 | around all of this stuff all divided by two A | |
| 02:48 | . So we have to kind of mirror image solutions | |
| 02:51 | . Notice everything is exactly the same , but one | |
| 02:53 | has a plus sign and one of them has a | |
| 02:55 | minus sign . We've been doing this over and over | |
| 02:58 | and over again for many , many , many lessons | |
| 03:00 | . So now what I'm trying to tell you is | |
| 03:02 | of course we can take A and B . And | |
| 03:03 | C . And we can stick it in and get | |
| 03:04 | these solutions , but we want to understand some characteristics | |
| 03:08 | of the solutions without actually calculating the entire exact answer | |
| 03:13 | . So in order to do that I'm going we're | |
| 03:15 | gonna define something , we're gonna say let capital D | |
| 03:19 | we're going to call it the discriminate , but we're | |
| 03:21 | going to just let it equal to whatever this stuff | |
| 03:23 | is under the radical B squared minus for a C | |
| 03:27 | . And this is called this triple thing means it's | |
| 03:29 | equal by definition to be what we call the discriminate | |
| 03:32 | the scrim in that . So when you really think | |
| 03:38 | about it , um the reason we're setting it equal | |
| 03:41 | to say , well why do we set it equal | |
| 03:42 | to what the stuff is under the radical ? It's | |
| 03:44 | because what is under this radical really governs what the | |
| 03:48 | solutions of the quadratic formula will be . When you | |
| 03:50 | think about it , that radical is the most important | |
| 03:54 | part of the whole thing really . Because if you | |
| 03:56 | think about it if what is under the radical And | |
| 03:58 | I mean when I say what is under it , | |
| 03:59 | I mean the whole thing , if it's a positive | |
| 04:02 | number then you'll be able to take the square root | |
| 04:04 | of it and you'll get real answers . But if | |
| 04:07 | what is under that radical actually turns out to be | |
| 04:10 | negative , then when you take the square root of | |
| 04:12 | it you're going to get imaginary numbers that means your | |
| 04:14 | answers will be imaginary . And if the what is | |
| 04:18 | under this radical is neither negative nor positive but actually | |
| 04:21 | exactly equal to zero . We have another special case | |
| 04:24 | . So we know that what is under this radical | |
| 04:26 | is really really important to tell us what the solutions | |
| 04:29 | look like . And we have a special name for | |
| 04:31 | that . What is under the radical here is called | |
| 04:33 | the discriminate . So any time that you're in an | |
| 04:35 | exam or a test or in class or a book | |
| 04:38 | you're reading or something that talks about the discriminate in | |
| 04:40 | the in your mind . You need to think that's | |
| 04:42 | just whatever the under the radical and the quadratic formula | |
| 04:45 | . That's what we called indiscriminate . So let me | |
| 04:48 | write it down a little bit more explicitly for you | |
| 04:50 | . These solutions here that we all know and love | |
| 04:53 | . We can rewrite them in terms of discriminate really | |
| 04:55 | easily . So that would be negative B plus the | |
| 04:57 | square root of this discriminate D . Because it's equal | |
| 05:00 | to the whole thing here . Over to a and | |
| 05:04 | the other solution X two is gonna be negative B | |
| 05:07 | minus the square root of this discriminate uh over to | |
| 05:12 | A . So you see what I mean here what | |
| 05:14 | is under this radical ? The special thing we call | |
| 05:16 | the discriminate governs the whole thing . It governs whether | |
| 05:19 | the solutions are real numbers , whether the solutions are | |
| 05:22 | complex uh answers . And that's what we mean by | |
| 05:26 | when we look at solutions of quadratic sits that's the | |
| 05:28 | most important thing about the roots is are they real | |
| 05:31 | numbers or are they imaginary numbers ? And in a | |
| 05:33 | few lessons I'm going to talk to you and a | |
| 05:35 | whole lot more detail about why we even get imaginary | |
| 05:37 | roots to begin with . But for now let's focus | |
| 05:40 | on what this discriminate is telling us . So there's | |
| 05:43 | really a couple of cases and what I'm gonna do | |
| 05:45 | is I'm gonna show you what happens when this D | |
| 05:47 | here this discriminate is different cases . We're gonna talk | |
| 05:50 | about it and then we're gonna go to the computer | |
| 05:52 | where we can graph and see exactly what's really happening | |
| 05:54 | here . So we're gonna assume here when we talk | |
| 05:58 | about the discriminate here that A and B and C | |
| 06:02 | are real . And what I mean by a B | |
| 06:04 | and C . I mean the coefficients in front of | |
| 06:06 | the other parts of my polynomial . They're real numbers | |
| 06:09 | . They can be negative positive fractions , decimals , | |
| 06:12 | square roots , that's fine . But they can't be | |
| 06:14 | I because it confuses things and clutters things up . | |
| 06:18 | If you talk about these things up here being imaginary | |
| 06:20 | . So for now let's just assume that they are | |
| 06:23 | real numbers . Then we have basically three main cases | |
| 06:27 | in the first case is if this discriminate which means | |
| 06:31 | whatever is under the radical and the quadratic formula is | |
| 06:34 | greater than zero means it's a positive number . Then | |
| 06:37 | what this means is that when I take the square | |
| 06:39 | root of a positive number , I'm going to get | |
| 06:41 | a positive answer . Right square root of 49 square | |
| 06:45 | root of 38 . I'm gonna get some kind of | |
| 06:46 | positive answer . I can then add to negative b | |
| 06:49 | , divide by this . But no matter what happens | |
| 06:52 | if this discriminates bigger than zero , I'm going to | |
| 06:54 | get a number that's going to be added or subtracted | |
| 06:56 | and so on . I'm going to get to roots | |
| 06:59 | as the answer . They're going to be both real | |
| 07:02 | words , no complex numbers anywhere and they're gonna be | |
| 07:05 | unequal . And this should make complete sense because for | |
| 07:09 | instance , if Ds were saying it's positive , let's | |
| 07:11 | say D becomes 36 then I take the square root | |
| 07:14 | , I'm gonna get a six , then I'm gonna | |
| 07:16 | basically add a negative B and divide by two A | |
| 07:19 | . But A and B A real number . So | |
| 07:20 | no matter what happens , I will always get a | |
| 07:22 | real answer . Because what came out of this radical | |
| 07:25 | was real because I'm adding in one case and subtracting | |
| 07:28 | in the other case I'm going to get unequal roots | |
| 07:31 | and of course I'm gonna get to routes because if | |
| 07:33 | I have the same exact thing here , I've got | |
| 07:35 | the same thing here . In one case I'm adding | |
| 07:37 | this number . In another case I'm subtracting a number | |
| 07:40 | then the roots are not going to be the same | |
| 07:42 | . That's what happens most of the time . In | |
| 07:43 | the quadratic formula , whenever you get a positive discriminate | |
| 07:46 | , you're gonna have two routes . One of them | |
| 07:49 | is going to be a little bit displaced from the | |
| 07:50 | other because it's plus and minus . But they're both | |
| 07:53 | going to be real . That's case number one case | |
| 07:57 | number two is you might guess . Well what happens | |
| 07:59 | if this discriminate actually isn't bigger than zero ? What | |
| 08:01 | if it's smaller than zero Then ? If you think | |
| 08:04 | through the logic let's say it's negative 36 you have | |
| 08:07 | under this radical . Then if you take the square | |
| 08:09 | root of that , you're gonna get six . I | |
| 08:11 | because you have a negative number under the radical , | |
| 08:13 | you always have to have an eye . So because | |
| 08:15 | of that you're still going to have to roots but | |
| 08:17 | they're not going to be real anymore because you've introduced | |
| 08:20 | an eye anytime this discriminate is less than zero . | |
| 08:23 | So you're gonna have to roots same as before . | |
| 08:27 | Um But they're going to be complex congregants . And | |
| 08:35 | remember complex conjugate means a complex conjugate is like one | |
| 08:39 | plus two I and one minus two I . Or | |
| 08:42 | one plus seven I one minus seven . I It's | |
| 08:45 | the same thing . You just have a negative sign | |
| 08:46 | in front of the imaginary part . You switch the | |
| 08:48 | sign of imaginary part . Those are conjugated . So | |
| 08:50 | if I have a negative 49 here , in a | |
| 08:53 | negative 49 here from id from my discriminate it . | |
| 08:58 | When I take the square root of that I'll have | |
| 08:59 | seven . I then I'll be adding seven I . | |
| 09:02 | And subtracting seven eyes . So they'll be complex conjure | |
| 09:05 | gets because I'm I'm switching the signs of the imaginary | |
| 09:07 | parts . I'll still have two of them two routes | |
| 09:10 | . Because in one case I'm adding , in one | |
| 09:11 | case I'm subtracting and that's what happens . Oftentimes with | |
| 09:14 | the quadratic formula you will get to routes that are | |
| 09:17 | complex conflicts of each other . Now the third case | |
| 09:21 | is the very special case . What if this discriminate | |
| 09:24 | exactly happens to equal zero ? What's going to happen | |
| 09:27 | there ? Well , it can happen because B squared | |
| 09:30 | minus four A . C . They're just , it's | |
| 09:31 | just a calculation based on my polynomial . So if | |
| 09:34 | I pick B and a . And C perfectly , | |
| 09:36 | I can make this thing go to zero , B | |
| 09:38 | squared minus four A . C . And if this | |
| 09:41 | D become zero then it's negative B plus square root | |
| 09:44 | of zero . But that just becomes zero . So | |
| 09:46 | it's basically negative B over two A . For the | |
| 09:49 | route number one . And if the discriminate here becomes | |
| 09:52 | zero then it's negative B minus again square root of | |
| 09:55 | zero . So zero negative B minus zero is nothing | |
| 09:58 | . So it's negative B over two A . For | |
| 09:59 | this and negative B over two A . For this | |
| 10:01 | . So if the discriminate is exactly equal to zero | |
| 10:05 | , I have real roots . All right . Because | |
| 10:07 | I haven't introduced any complex numbers but it's called double | |
| 10:10 | roots . So , you know occasionally we talked about | |
| 10:15 | double roots right ? Where we have the the parabola | |
| 10:19 | just kissing the X axis , just touching tangentially the | |
| 10:22 | X axis . And we say there's two routes right | |
| 10:24 | on top of each other right where it touches . | |
| 10:25 | Well that's because in the quadratic formula you have a | |
| 10:28 | square root of zero here . So the two routes | |
| 10:31 | end up becoming the same thing negative B . Over | |
| 10:33 | two A . You technically have two of them but | |
| 10:35 | they're the exact same thing and that happens when they | |
| 10:38 | discriminate is equal to zero . So what I want | |
| 10:41 | you to do is keep this in the back of | |
| 10:42 | your mind and the the discriminate here as we all | |
| 10:46 | know , we're going to write it down here . | |
| 10:47 | The discriminate is equal by definition to be b squared | |
| 10:50 | minus four at times . Hmc The square root is | |
| 10:53 | not part of the discriminate . It's just what's underneath | |
| 10:56 | the square root . That's what we talk about what | |
| 10:58 | the discriminate really is . So there's three cases if | |
| 11:01 | the discriminates bigger than zero . we know that we | |
| 11:03 | have two routes . They're real roots in their unequal | |
| 11:06 | because we have plus and minus , whatever's under that | |
| 11:08 | radical after you take the square root . If it's | |
| 11:10 | less than zero we still have two routes . But | |
| 11:12 | the complex country gets because we have negative numbers under | |
| 11:14 | the square roots . If the discriminates actually equal to | |
| 11:17 | zero perfectly , then we've added and subtracting zero . | |
| 11:21 | So we have exactly the same routes two times over | |
| 11:24 | . So we call it real roots because there's no | |
| 11:26 | eyes involved and it's a double route . Now . | |
| 11:28 | What I want to do now is I could just | |
| 11:29 | leave it here and say let's solve some problems , | |
| 11:31 | but I really think it's uh instructive to go look | |
| 11:34 | at what happens . So let's go over to the | |
| 11:36 | computer and take a look at what happens when we | |
| 11:37 | look at the discriminate of different kinds of quadratic equations | |
| 11:42 | . Okay , welcome back in this case we have | |
| 11:44 | , what we have here in this demo here is | |
| 11:46 | we have initially we have X squared which is a | |
| 11:49 | parabola that you all know and love . But with | |
| 11:51 | these sliders here I can actually change what this parabola | |
| 11:54 | looks like . And what we're gonna do over here | |
| 11:56 | is take a look at this is the quadratic formula | |
| 11:58 | negative B plus or minus square B squared minus four | |
| 12:01 | ac over to a But here I've calculated the discriminate | |
| 12:05 | which is now we calculate to be zero . So | |
| 12:08 | that was that very special case . Remember when the | |
| 12:10 | discriminates equal to zero ? The roots here which are | |
| 12:12 | also being calculated is the double room located at zero | |
| 12:15 | which is exactly what the graph shows because this parabola | |
| 12:19 | kisses and just touches the the X axis here , | |
| 12:21 | right at X is equal to zero . So notice | |
| 12:24 | if you calculate this discriminate B squared minus four . | |
| 12:27 | A . C . A . Is gonna be one | |
| 12:30 | because it's one X squared B a zero and see | |
| 12:32 | a zero . So B squared minus four A . | |
| 12:35 | C . When you work it out , be square | |
| 12:36 | would be zero minus four times a times C a | |
| 12:39 | zero in this equation . So it's basically discriminate zero | |
| 12:42 | and that's what we have here . Now let's take | |
| 12:44 | a look at a couple of different cases , let's | |
| 12:46 | make the discriminate positive . We can do that easily | |
| 12:49 | by shifting this graph up right ? So we have | |
| 12:51 | X squared plus one . Now the discriminate is actually | |
| 12:55 | negative . Let's go in and I actually went the | |
| 12:57 | wrong way let's let's go and first make it positive | |
| 12:59 | . Let's go down this direction . And you can | |
| 13:01 | see that what's going on here is now the equation | |
| 13:03 | is x squared minus two . If you calculate b | |
| 13:06 | squared minus four A C it'll be negative eight because | |
| 13:11 | B is zero , so it's zero minus . And | |
| 13:14 | then you have four times a times c four times | |
| 13:17 | one here times negative two . Uh In the minus | |
| 13:20 | sign here negative times negative positive . You actually get | |
| 13:23 | a positive discriminate for this case . So because you | |
| 13:26 | have a positive discriminate , that's the case . When | |
| 13:27 | you have to real roots in this case you see | |
| 13:30 | the roots are calculated negative 1.4 positive 1.4 . And | |
| 13:35 | so when the discriminate is positive we get to two | |
| 13:37 | real roots which are separated like this . Now I | |
| 13:39 | can change this equation slightly now this is a very | |
| 13:42 | slightly different one . But you can see again the | |
| 13:44 | discriminate is positive and we get to real roots , | |
| 13:47 | I can flip it around uh make it make it | |
| 13:49 | a little more narrow , we still have two crossing | |
| 13:51 | points , we still have to real roots here . | |
| 13:53 | There's no imaginary numbers involved . Indiscriminate is positive . | |
| 13:56 | So no matter how I actually jockey this thing around | |
| 13:59 | I can bring it over on this side , I | |
| 14:01 | can change it , make it more like this , | |
| 14:03 | let me shift it up a little bit . Something | |
| 14:05 | like this . The discriminate again is positive . And | |
| 14:09 | then I see that I have the two real roots | |
| 14:11 | . So the way I want you to burn it | |
| 14:12 | in your mind is any time this discriminate turns out | |
| 14:14 | to be positive . The graph crosses the X . | |
| 14:17 | Axis in two locations and then you have the two | |
| 14:20 | routes which are calculated here . Now let's go and | |
| 14:22 | see what happens when the discriminates negative . So I | |
| 14:24 | can shift this graph up and take a look at | |
| 14:26 | what's going on in this parameter right here I have | |
| 14:29 | X squared minus four X plus five . Now when | |
| 14:32 | you work through that b squared minus four A . | |
| 14:34 | C . Discriminate you're going to get a negative four | |
| 14:36 | which means when I take the square root of the | |
| 14:38 | negative number , I'm gonna introduce imaginary numbers . So | |
| 14:42 | now my two routes are actually imaginary . So this | |
| 14:44 | covers the case when the discriminates negative you get the | |
| 14:47 | to imagine or too complex routes which are congregates of | |
| 14:50 | one another . Notice the two and the one I | |
| 14:53 | . And two plus one I two minus one . | |
| 14:54 | I it's exactly the same thing , differing only by | |
| 14:57 | a sign . And I can move this guy . | |
| 14:59 | Let me try to keep it upstairs up here , | |
| 15:01 | I can move this all over the place . Let | |
| 15:03 | me something shifted over like right over on this side | |
| 15:06 | of the graph here you can see I still have | |
| 15:08 | a negative four for indiscriminate . I still have complex | |
| 15:10 | conjugate roots now , interestingly I can change this , | |
| 15:14 | let me go and try to move it back to | |
| 15:16 | where I started here just so you can kind of | |
| 15:17 | see what goes on when I change this uh to | |
| 15:21 | larger and larger numbers it gets narrow but I can | |
| 15:23 | go the other direction so let me go and make | |
| 15:25 | this guy negative and bring it down like this so | |
| 15:28 | that you have an upside down parabola like this , | |
| 15:30 | there's still two crossing points . I still have a | |
| 15:33 | positive discriminate when you work out B squared minus four | |
| 15:36 | . A . C . Is still positive . And | |
| 15:37 | because of that I have my two real roots , | |
| 15:39 | I can move this upside down parabola wherever I want | |
| 15:42 | to , as long as my discriminate is positive , | |
| 15:45 | which you can see it is positive . In all | |
| 15:46 | of these cases I have my two routes which are | |
| 15:49 | real but as soon as I grab this guy and | |
| 15:53 | bring it down below the axis . So I don't | |
| 15:54 | have any actual real crossing points to discriminate turns negative | |
| 15:58 | when you work out B squared minus four A . | |
| 16:00 | C . Here you're gonna get that negative discriminate . | |
| 16:02 | And whenever that happens no matter where I shift this | |
| 16:04 | thing when it's underneath the axis and I have that | |
| 16:06 | negative discriminate . Uh I'm gonna get these complex conjugate | |
| 16:09 | roots . So that's what I want you to understand | |
| 16:11 | . When the discriminates positive , you get real roots | |
| 16:14 | and when the discriminates negative you get these complex conjugate | |
| 16:16 | roots . Now what we need to do is talk | |
| 16:18 | about the very special case when the discriminate is equal | |
| 16:21 | to zero . So let me flip this back around | |
| 16:24 | and bring it back to where we started this thing | |
| 16:26 | from . Uh Because that's the best way to start | |
| 16:29 | here . And we had that very special case with | |
| 16:31 | the quadratic Y . Is equal to x squared when | |
| 16:33 | the discriminate actually already equal zero . And that was | |
| 16:36 | the third case when we had the discriminate equal zero | |
| 16:39 | . We talked about on the board how that gives | |
| 16:41 | us basically two identical roots . In this case they're | |
| 16:43 | both centered here at X is equal to zero . | |
| 16:46 | Now I want to dial in a couple of different | |
| 16:47 | equations with double roots . So let's do X squared | |
| 16:50 | , let's do minus two X plus one . I've | |
| 16:53 | already figured this out ahead of time . So here's | |
| 16:55 | another quadratic equation . You can see the graph goes | |
| 16:58 | down and touches the access and only one location . | |
| 17:01 | So we expect a double root right here at X | |
| 17:03 | is equal to one . So we have two routes | |
| 17:05 | , X is equal to one identical roots and the | |
| 17:07 | discriminate for the zero . So if you dial in | |
| 17:10 | B squared minus four A . C . And calculate | |
| 17:12 | it you're gonna get a zero . That's why we | |
| 17:14 | have the double roots located here . Let me show | |
| 17:17 | you 141 more . We'll do negative four X squared | |
| 17:21 | um Plus 12 X . And see if I can | |
| 17:23 | get all the way to 12 X . Like this | |
| 17:25 | . Uh what's going to go up a little bit | |
| 17:27 | more to 12 X . And then we're gonna go | |
| 17:28 | minus nine . I've already looked at this ahead of | |
| 17:30 | time so I know exactly where to go . It's | |
| 17:32 | kind of hard to find them randomly . But if | |
| 17:34 | you if you run B squared minus four A . | |
| 17:36 | C . Through this discriminate here A being negative for | |
| 17:38 | be being 12 and C being negative nine Then you'll | |
| 17:42 | get to discriminate exactly at zero and you'll see that | |
| 17:44 | you have a double root here exactly at 1.52 routes | |
| 17:47 | in the same location . So the bottom line most | |
| 17:50 | important thing for you to understand in all of this | |
| 17:53 | stuff . The only reason I even put this together | |
| 17:54 | is I really like visual things when I can do | |
| 17:57 | it without too much , you know , too much | |
| 17:59 | work . And I think it's really really nice for | |
| 18:01 | you to see that when these graphs , when these | |
| 18:03 | quadratic graphs cross the X axis and two locations that | |
| 18:07 | we have to real roots and you can see them | |
| 18:09 | here and no matter where I move this thing wherever | |
| 18:12 | I shifted , as long as I have to real | |
| 18:14 | crossing points , I'm going to have the two real | |
| 18:16 | roots in the discriminate will be positive . But as | |
| 18:18 | soon as this graph pops up above where there's no | |
| 18:20 | crossing points on the X axis . Then at that | |
| 18:23 | point the discriminate always turns negative . And I don't | |
| 18:25 | have any real roots anymore because I don't have any | |
| 18:27 | real crossing points . I have these imaginary or complex | |
| 18:31 | conjugate roots . And then of course you have the | |
| 18:33 | very special case which you just looked at a minute | |
| 18:35 | ago of basically whenever you don't have um let me | |
| 18:40 | go and get it over here when you don't have | |
| 18:42 | uh two distinct crossing points you have basically the graph | |
| 18:45 | just touches tangentially the axis . The simplest cases why | |
| 18:49 | is equal to X squared in that case of discriminates | |
| 18:51 | not not positive , that discriminates not negative . But | |
| 18:54 | in the case of double routes to discriminate turns out | |
| 18:56 | to exactly be right in the middle of those at | |
| 18:58 | zero . In that case we have two routes and | |
| 19:01 | in fact you can see the roots here , the | |
| 19:03 | roots are really far apart , negative two point to | |
| 19:06 | positive 2.2 . And as we get closer and closer | |
| 19:09 | , the roots get closer and closer together , closer | |
| 19:11 | and closer together , closer and closer together . Finally | |
| 19:13 | the roots are exactly on top of each other . | |
| 19:15 | We have two of them . And then now we | |
| 19:16 | don't even have any real roots at all . They're | |
| 19:18 | just complex numbers . And we're gonna talk more about | |
| 19:20 | that a little bit later . So for now I'm | |
| 19:22 | gonna close this section out , we're gonna work some | |
| 19:24 | problems uh in the next lesson where we talk about | |
| 19:26 | this a little bit more concretely . All right , | |
| 19:30 | So now that we have done the computer work , | |
| 19:32 | we have solidified what we wanted to talk about , | |
| 19:34 | which is how to discriminate , predicts what the roots | |
| 19:37 | will look like . If the discriminates positive . We're | |
| 19:40 | gonna have to routes which are real . We saw | |
| 19:42 | that many times over . If the discriminates negative we're | |
| 19:44 | gonna have to routes which are complex and the consequence | |
| 19:47 | of each other . If the discriminate is exactly equal | |
| 19:49 | to zero then we're gonna have a double room where | |
| 19:52 | the graph is just touching the X . Axis and | |
| 19:54 | in one location really but it counts two times and | |
| 19:56 | that's what we get in the case of a real | |
| 19:58 | double route . Now there's one more thing I want | |
| 20:00 | to show you before we go on and do the | |
| 20:02 | problems in the next lesson and that is kind of | |
| 20:04 | like something that's still related to the discriminate but I | |
| 20:06 | wanted to save it for the end . It just | |
| 20:08 | makes a little more sense to put it there after | |
| 20:09 | the demo , what I want to talk to you | |
| 20:12 | about is what happens if what is under here ? | |
| 20:16 | The discriminate here . What happens if it's a perfect | |
| 20:18 | square ? In other words we can take the square | |
| 20:21 | root of 13 . Of course we have a decimal | |
| 20:23 | answer but the square root of 13 is a is | |
| 20:26 | an infinite decimal . It's it's not , you can't | |
| 20:28 | really take the decimal value of it unless you truncate | |
| 20:30 | it , the square root of two , the square | |
| 20:32 | root of three , they go on and on forever | |
| 20:33 | when you convert to decimals . So , but some | |
| 20:36 | square roots are perfect squares , like square root of | |
| 20:39 | 36 . We know that that's six . Exacting square | |
| 20:42 | root of 49 is seven . We know that that's | |
| 20:44 | exactly the case . We know that the square root | |
| 20:47 | of 110 . We know that some of these things | |
| 20:49 | we call perfect squares when we have a perfect square | |
| 20:52 | that lives under there . Taking the square root produces | |
| 20:54 | a lot more simple answer . We don't have any | |
| 20:56 | radicals in those answers . So let's talk about the | |
| 20:59 | case when we have a perfect square under there . | |
| 21:02 | So first let me say if we have two conditions | |
| 21:05 | , the first condition is we have integral coefficients of | |
| 21:13 | my quadratic equations . In other words , A and | |
| 21:15 | B and C . Or they're not decimals are not | |
| 21:17 | fractions , they're not in there . Just whole numbers | |
| 21:19 | negative or positive . That's what an integral coefficient means | |
| 21:22 | . Also , it means we could we could transform | |
| 21:24 | our original quadratic equation by multiplication or something into something | |
| 21:28 | with the interval coefficients . The other constraint is let's | |
| 21:31 | see what happens if this discriminate is is equal to | |
| 21:34 | a perfect square , which is rare actually because there's | |
| 21:41 | only so many perfect squares . I mean four is | |
| 21:43 | a perfect square nine . Perfect square 16 , 25 | |
| 21:47 | . You know , you can go up 36 so | |
| 21:49 | on . Those are the special ones where you can | |
| 21:51 | take the square root and you get just a whole | |
| 21:52 | number back . Right . Very special numbers . So | |
| 21:55 | if you have a quadratic equation with integral coefficients and | |
| 21:58 | if the discriminate is one of these very very special | |
| 22:01 | perfect numbers like 25 36 things like that then what | |
| 22:05 | you're going to have , when you calculate this thing | |
| 22:08 | , the solutions is gonna be negative B plus or | |
| 22:11 | minus the square root of this . Perfect square All | |
| 22:18 | divided by two a . So b squared minus four | |
| 22:21 | A . C . Is just a perfect number , | |
| 22:22 | like 36 or 100 or 49 or something like that | |
| 22:26 | . Um but not a perfect square . Those very | |
| 22:30 | special numbers . All other numbers are not what we | |
| 22:32 | consider perfect squares . So if you have a perfect | |
| 22:35 | square that lives under here then what you're gonna get | |
| 22:37 | is an answer is going to be rational roots . | |
| 22:42 | Right ? Remember the word rational . We talked about | |
| 22:46 | that before . The word rational just means you can | |
| 22:48 | write it as a fraction . So in other words | |
| 22:50 | , if no matter what B is an A . | |
| 22:53 | Is if what is under here is a perfect square | |
| 22:55 | . Like let's say it's 36 I'll take the square | |
| 22:57 | root , it'll give me a six , then negative | |
| 22:59 | B plus six over to A . If A B | |
| 23:02 | and C are all integral coefficients , you see then | |
| 23:05 | what if my radical basically disappears because I can take | |
| 23:08 | the square root of it and get that number back | |
| 23:10 | then no matter what I'm adding or subtracting or dividing | |
| 23:13 | by , it will be rational . The answer will | |
| 23:15 | be rational because it will be a fraction . In | |
| 23:17 | other words , there will be no radicals and the | |
| 23:23 | answer . So for instance , to contrast it , | |
| 23:26 | if I put 13 under the radical , then it | |
| 23:28 | would be negatively whatever that is plus the square root | |
| 23:31 | of 13 over to a whatever it is , I | |
| 23:34 | cannot get rid of that square root of 13 . | |
| 23:36 | Thirteen's irrational . It goes on and on and on | |
| 23:38 | forever as a decimal . When you try to convert | |
| 23:40 | it to a decimal . So the last part of | |
| 23:43 | this lesson is just telling you that hey , There's | |
| 23:46 | certain constraints on D that tell us a lot about | |
| 23:49 | the solutions . If it's positive , we have two | |
| 23:51 | routes that are real . If it's negative , we | |
| 23:52 | have two routes that are complex . If it's zero | |
| 23:55 | , we have two routes that are real and also | |
| 23:57 | what we call a double route and on top of | |
| 23:59 | that , if what is under here is a very | |
| 24:00 | special number like 36 or 49 or 100 or nine | |
| 24:04 | or 16 or something like that . Then when you | |
| 24:07 | do all the math here , you're always going to | |
| 24:08 | get a rational answer like one half or one third | |
| 24:12 | or 22 meaning you can write it as a fraction | |
| 24:15 | 22/1 no radicals in the final answer because I've gotten | |
| 24:19 | rid of my square root when I have that perfect | |
| 24:20 | square there . But if I have something other than | |
| 24:22 | a perfect square , they're like 20 Well I can't | |
| 24:25 | get rid of that square root square root of 20 | |
| 24:27 | , I can do a factor tree but I'm still | |
| 24:29 | going to have a radical left . So the answer | |
| 24:31 | will not be rational . So sometimes on your test | |
| 24:33 | you might say , hey here's an equation . Don't | |
| 24:36 | use the quadratic formula , but just calculate the discriminate | |
| 24:39 | and tell me if the answers are gonna be rational | |
| 24:41 | or not . All you have to do is say | |
| 24:43 | , well what's under this radical ? Is it a | |
| 24:45 | perfect square or not ? And that will tell you | |
| 24:47 | if it is a perfect square , you're going to | |
| 24:48 | get rational answers , meaning you can you can have | |
| 24:51 | no radicals in your answers , so make sure I | |
| 24:53 | understand this . Follow me on to the next lesson | |
| 24:55 | . We're going to use everything we've learned here in | |
| 24:56 | order to actually solve some problems involving the discriminate . |
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