01 - Visualize Roots of Equations - Linear, Quadratic, Cubic, Quartic Solutions - By Math and Science
Transcript
00:00 | Hello . Welcome back to the Algebra two course . | |
00:02 | This is Unit nine . The topic of this batch | |
00:05 | of lessons is basically going to be zeroing in on | |
00:07 | solving these quadratic equations . You can also call it | |
00:11 | finding the roots of these quadratic equations . But what | |
00:14 | we're gonna do in this lesson is we're going to | |
00:15 | back up the truck a little bit , zoom out | |
00:17 | a little bit so that we don't forget what we're | |
00:19 | doing . And I have a computer demo that I'm | |
00:21 | gonna do for you here in about 20 seconds . | |
00:23 | That's going to hopefully illuminate a lot of what we're | |
00:25 | talking about before we get back to the board and | |
00:27 | write the math down . So remember the point is | |
00:30 | when we're talking about roots of an equation , What | |
00:32 | we're really doing is we set that equation equal to | |
00:34 | zero . Because those will be the points where the | |
00:37 | graph crosses the X axis . Because those are the | |
00:39 | points where the function is equal to zero . The | |
00:41 | function is the y axis . Right ? So if | |
00:43 | you set it equal to zero , you're finding all | |
00:45 | the points where it crosses that X axis where the | |
00:47 | value is equal to zero . So we have lines | |
00:50 | which are linear guys like this that only crossed the | |
00:53 | X axis in one location . We have quadratic , | |
00:56 | which look like the parameters we've talked about . They | |
00:58 | generally cross the axis in two locations and then we | |
01:00 | have Q bix and then we have the higher order | |
01:03 | polynomial is beyond that . So what we're going to | |
01:05 | be doing here is covering the general idea of linear | |
01:08 | equations , quadratic cubic , which is the power of | |
01:11 | three and the court equations , which is the power | |
01:14 | of four . So before we get back to the | |
01:16 | board here , let's do that . Computer demo . | |
01:17 | And I'll show you graphically what the roots of these | |
01:19 | different kinds of equations really look like . Okay , | |
01:23 | welcome back here . What we have here is a | |
01:25 | little demo that I've done here , where I can | |
01:27 | show you the different the graphs of the different kinds | |
01:30 | of equations here . We have linear equations and the | |
01:32 | higher order polynomial beyond that . So the simplest kind | |
01:35 | of equation you can have is a horizontal line . | |
01:38 | So the graph here , it just has three here | |
01:40 | . That means why equals three . That's a horizontal | |
01:42 | line that crosses it at the value of Y is | |
01:45 | equal to three here . The line crosses at the | |
01:47 | value of Y is equal to four . You can | |
01:49 | see that this horizontal line does not cross the X | |
01:51 | access at the X axis at all . So it | |
01:53 | doesn't have any roots . But if we make the | |
01:56 | line a little more interesting , if we put a | |
01:58 | slant to it , then here's the equation of this | |
02:00 | line two X plus four . You can see the | |
02:02 | y intercept is up here at four and it has | |
02:05 | a slope of to rise over run , that's what | |
02:07 | this means , right . And you can see that | |
02:08 | in general winning line is slanted , it is going | |
02:11 | across the X axis in one location . Now I | |
02:13 | can play around with this line , I can bring | |
02:16 | it down like this . So the y intercept is | |
02:17 | negative three down here and you can see that the | |
02:20 | root here of this first order equation . See the | |
02:23 | power of X here is just a one . So | |
02:25 | we have a first order route here which basically means | |
02:29 | we expect to have one solution when we set this | |
02:32 | guy equal to zero , which means one crossing point | |
02:34 | . Here's the crossing point . Now , no matter | |
02:36 | what I do , even if I make the lines | |
02:38 | slant the other direction . If I shift it up | |
02:41 | or down , basically move the line around , it's | |
02:43 | only going across the X axis in one location has | |
02:45 | one root because the power of X is just a | |
02:48 | number one . Now let's go up . And I'm | |
02:50 | gonna drag my slider here to make a quadratic . | |
02:53 | So this is a simple quadratic and I can reset | |
02:55 | the rest of these guys here to kind of reset | |
02:57 | what I have here . So this is the quadratic | |
02:59 | , Y is equal to x squared If I change | |
03:02 | it to make it for instance , X squared minus | |
03:04 | three . This is what this curve looks like now | |
03:06 | the highest power of X here is a two . | |
03:08 | That means it's a quadratic . All quadratic have a | |
03:10 | power of two for the highest power . And because | |
03:13 | the power is too , that means that we have | |
03:15 | to roots of this equation right ? In this case | |
03:18 | one route is where it crosses the X axis here | |
03:21 | , and the other route is where it crosses the | |
03:23 | axis over here . So you can see that there | |
03:24 | are mirror images . Now , as I change the | |
03:26 | graph a little bit , you can see the crossing | |
03:28 | points are in different locations . But there's still two | |
03:31 | routes . Now , if I change very slightly what | |
03:33 | the equation looks like , maybe make the problem or | |
03:36 | more sharp by putting a higher coefficient in front of | |
03:39 | the X squared here . It does change the shape | |
03:42 | of the curve , but we still have two crossing | |
03:44 | points . And with quadratic , sui have always expect | |
03:47 | to have two crossing points because the highest power of | |
03:49 | X is a two , just like in the case | |
03:51 | of a line like this line right here or any | |
03:55 | line in general , we always expect only one crossing | |
03:57 | point because the highest power of X is one here | |
03:59 | . So let's go back up and make it a | |
04:00 | quadratic . Now you can see we have two crossing | |
04:03 | points . You can see there's a crossing point here | |
04:05 | , there's a crossing point here . So we have | |
04:06 | two routes . Now what I want to draw your | |
04:08 | attention to is if I drag this quadratic above the | |
04:11 | X axis so that you see how it never intersects | |
04:14 | this quadratic never intersects the X . Axis at all | |
04:16 | . That means that this function never ever has a | |
04:19 | value where it's equal to zero . So it doesn't | |
04:22 | have any real roots . But as we're gonna learn | |
04:25 | uh here very soon when we talk about the quadratic | |
04:27 | formula , we're going to see that this quadratic equation | |
04:30 | still has two roots , but because we don't have | |
04:33 | any crossings on the X axis , they're not real | |
04:35 | roots . We always have two routes , but it's | |
04:37 | going to turn , we're going to turn out and | |
04:38 | see that when the graph is floating up here , | |
04:41 | we have two complex roots are imaginary roots . We've | |
04:44 | talked about complex numbers before . So this quadratic equation | |
04:47 | still has two roots . It's just that they're both | |
04:49 | complex and so we can't see them on this graph | |
04:52 | of real numbers , the X and Y graph here | |
04:54 | . So you see , no matter what I do | |
04:56 | , if I move this guy maybe over here , | |
04:58 | maybe shift away up high . You see down below | |
05:00 | once it crosses down below , we have two real | |
05:03 | roots as I increase it , we still have to | |
05:05 | real roots , we still have to real roots . | |
05:07 | They're getting closer and closer and closer together . We | |
05:09 | have two real roots right here here , we still | |
05:12 | have to real roots , here's one here and here's | |
05:14 | one here . But as I pop over to the | |
05:15 | other side like this , then I have actually no | |
05:19 | real roots , but I still have two routes in | |
05:21 | total . I have to complex roots and we're going | |
05:23 | to learn how to calculate the complex roots with the | |
05:26 | quadratic formula here very soon . All right . One | |
05:29 | more thing I'm gonna say before I leave quadratic here | |
05:32 | is let's do our basic quadratic that's centered on the | |
05:34 | X axis right ? Here are centered on the like | |
05:37 | this guy that the general quadratic like two X squared | |
05:39 | . Or you could say x squared notice here that | |
05:42 | as I uh let me make it a simpler one | |
05:44 | . Let's make make it why is equal to X | |
05:47 | squared ? So here's the general y is equal to | |
05:49 | X squared graph . Right ? Let me drag it | |
05:51 | down below . And you can see that there's two | |
05:53 | routes , one here and one here . Now as | |
05:56 | I move it up higher , let's go farther down | |
05:58 | . You see the roots are very far spaced apart | |
06:00 | because the thing is way below the X . Axis | |
06:03 | as I make this and shift this graph up the | |
06:05 | roots get closer and closer together . You see what's | |
06:08 | happening . They're getting closer here is negative one and | |
06:10 | positive one . And as eventually I get to make | |
06:13 | the graft shift up to this point , the roots | |
06:16 | have gotten so close together that they're right on top | |
06:18 | of each other . The roots were actually right here | |
06:20 | at X . Is equal to zero but there's still | |
06:22 | two of them . It's a quadratic . There has | |
06:24 | to be two routes , they're right on top of | |
06:26 | each other and we call that a double route . | |
06:28 | So it's easy to see when you actually start down | |
06:30 | here . We have double roots , double roots , | |
06:32 | double roots , double I'm sorry not double roots . | |
06:34 | We have two routes to roots , two routes to | |
06:36 | roots and so on . They're getting closer . Eventually | |
06:39 | we still have to routes that are right on top | |
06:40 | of each other , we call that a double route | |
06:42 | and then as we go above the X axis we | |
06:44 | have don't have any real roots at all , no | |
06:46 | crossing points , but we still have to complex roots | |
06:49 | which will learn about later . Now let's increase it | |
06:52 | to a cubic , This is what a general cubic | |
06:54 | function looks like and I can play with it to | |
06:56 | try to make it more like a , like a | |
06:58 | snake , it's kind of like a snaky function . | |
07:00 | And so this is what a cubic looks like . | |
07:03 | And you can see here have the highest power of | |
07:05 | excess of three . So we expect three roots of | |
07:08 | this equation , three crossing points . So here's a | |
07:10 | crossing point of the X axis , here's another crossing | |
07:12 | point here and here's another one . So there's three | |
07:14 | roots of this equation in general . All cubic functions | |
07:17 | are gonna look like this or they look like a | |
07:19 | mirror image flip down below . In other words , | |
07:21 | it can go up and down or down and up | |
07:23 | , but it'll still mostly look like a snaky kind | |
07:25 | of function like this . And as I go and | |
07:28 | drag this guy down interestingly , there's only one crossing | |
07:32 | here of this function here . It still looks like | |
07:33 | a like a snake , but we don't have one | |
07:35 | root , which means one real root , we can | |
07:37 | see over here , but you have to remember the | |
07:40 | highest power here is a three . We always have | |
07:42 | to have three roots total . So here we have | |
07:45 | one actual crossing point and we have two other routes | |
07:48 | which you could find . We're going to learn how | |
07:50 | to solve these cubic slater and you'll find out you | |
07:53 | have to complex roots for a total of three to | |
07:55 | complex roots and then the one real root here . | |
07:58 | As I drag the function up , we still have | |
08:00 | to complex roots and then the one real root over | |
08:03 | here . As I drag it up here , we | |
08:05 | have one real root here and notice we have a | |
08:08 | double room now because the function is now touching the | |
08:11 | X axis . And then as I go through it | |
08:14 | and we have three real roots , one , here's | |
08:16 | two and here's three . And as I shift the | |
08:18 | function up , we still have three real roots here | |
08:20 | , three crossing points . And then we have on | |
08:22 | this case we have the function crossing here . That's | |
08:24 | a real route . We have a double room now | |
08:26 | for a total of three routes here . And then | |
08:28 | as the function gets shifted up uh it's a little | |
08:31 | hard to see because it's shifted off the screen . | |
08:32 | That's probably better . We have now a one real | |
08:35 | root here on the negative side . And then we | |
08:38 | only have , we don't have any other crossing points | |
08:39 | . So the other two come up to make two | |
08:42 | more imaginary or complex roots . So the bottom line | |
08:45 | is you always have as many routes as you do | |
08:47 | the highest power . But some of them can be | |
08:49 | complex if you don't have enough crossing points in your | |
08:52 | graph and that's the what it means to get complex | |
08:56 | roots . Those are just routes that can't be displayed | |
08:58 | on a real graph like this . But the real | |
09:00 | crossing points will show up as roots as well . | |
09:03 | Now real quickly we're not gonna go through the cortex | |
09:05 | and higher too much . But let me just show | |
09:06 | you generally what in general cortex function looks like , | |
09:09 | you can play around with it , make it look | |
09:10 | various ways . So here in general you can see | |
09:13 | it looks like a W . So whereas the cubic | |
09:15 | look like a snake , the cortex look like a | |
09:17 | W . Of course it could be flipped upside down | |
09:19 | to look like an M . As well . But | |
09:21 | if I drag this function down here , then this | |
09:25 | core tech equation , this one here that we have | |
09:27 | printed at the top has to real roots because we | |
09:29 | have two crossing points . But we don't but we | |
09:32 | don't have any more actual crossing points . So the | |
09:34 | other two routes that we expect must be complex . | |
09:36 | So this graph would have to real roots and too | |
09:39 | complex routes that you can't see on this graph as | |
09:41 | I shifted up , it's the same deal . We | |
09:43 | have to real roots and too complex here . We | |
09:46 | have too real and too complex . But now you | |
09:48 | see the graph crosses and four locations . One , | |
09:51 | 23 and four . So this graph has actually four | |
09:54 | real roots and it doesn't have any complex roots . | |
09:57 | And as I shifted up , we have what's let's | |
10:00 | go here here . We have the same deal here | |
10:02 | . 1234 real roots and no complex roots . And | |
10:06 | as we shift , let me see if I can | |
10:07 | get it here and can I have the granularity . | |
10:09 | But as we go up higher above the axis of | |
10:11 | this graph , this equation doesn't have any crossing points | |
10:15 | at all . So all four of these routes must | |
10:18 | be complex . So you can kind of predict from | |
10:20 | the graph how many real and how many complex which | |
10:23 | you're going to have now , I'm just gonna fly | |
10:24 | through the rest of them because after you get to | |
10:27 | the cortex , the rest of them are just kind | |
10:28 | of hard to kind of visualize . But you can | |
10:30 | see 1/5 order equation still kind of looks like a | |
10:33 | snaky function . And if you uh if you push | |
10:37 | it down like here you can see you have now | |
10:39 | five crossing points , 1234 and then five matching the | |
10:44 | five routes that we expect here are the five real | |
10:46 | crossing points . If I shift this equation up , | |
10:49 | changing it slightly like this now the roots will be | |
10:52 | one real root and all four of the remaining two | |
10:55 | equal five must be complex roots and I can do | |
10:58 | the same thing by dragging it down here . This | |
10:59 | equation has one real root way over here and then | |
11:02 | all of the other four must be complex roots . | |
11:05 | And then in the middle as I shifted up and | |
11:07 | down here you can see you can have various combinations | |
11:09 | here . I have the 45 crossing points in total | |
11:12 | . And just playing around with it more . You | |
11:13 | can you can make crazier , higher order things like | |
11:16 | this . Me see if I can find something interesting | |
11:19 | . Uh Here you have one crossing point . Two | |
11:21 | crossing points . 34 So I only have four crossing | |
11:25 | points that I can see in this function . But | |
11:27 | we know that there must be six routes because it's | |
11:29 | 1/6 order polynomial . So we have four real . | |
11:32 | Therefore we must have too complex . So if you | |
11:34 | solve and find all of the roots of this equation | |
11:36 | , then you'll find four crossing points for rail routes | |
11:39 | and then the other two must be complex . And | |
11:41 | then we'll just do one more just because I know | |
11:43 | you're curious . The seventh order . See what I | |
11:45 | can make it look interesting . So this is the | |
11:47 | seventh order polynomial . You can see the graph of | |
11:49 | it , but how many crossing points do we have | |
11:50 | ? One ? 2345 We have five real roots , | |
11:55 | therefore the other two to make seven must be complex | |
11:58 | roots . So I wanted to show you the general | |
12:00 | idea of what we were doing so that you can | |
12:02 | get uh an idea of what real , what of | |
12:04 | what real roots are and what complex roots really represent | |
12:08 | . But what I want to do here closes , | |
12:09 | go back to the original thing . When we have | |
12:12 | our general general quadratic equation are famous . F . | |
12:15 | Of X is equal to X squared right ? If | |
12:18 | it has two crossing points , that's too real roots | |
12:21 | . If you have no crossing points at all , | |
12:23 | that's to purely complex or imaginary roots that you're gonna | |
12:26 | get it when you solve this guy . And then | |
12:28 | right , when it kisses the axis , you have | |
12:30 | to real roots , which we call a double route | |
12:32 | . You can see them getting closer and closer and | |
12:34 | closer together until now they're just basically right on top | |
12:36 | of each other . Uh fucking adjust it here , | |
12:39 | right here at X . Is equal to zero . | |
12:40 | So now let's go back to the board and wrap | |
12:42 | it up with writing down some equations . All right | |
12:47 | . So now that we've seen the demo of how | |
12:48 | these routes really behave when we have higher and higher | |
12:51 | order polynomial is what we want to do is kind | |
12:53 | of right . Some of the things down to make | |
12:54 | sure you understand and have it all in one place | |
12:57 | . So the simplest kind of equation that we really | |
12:59 | have and learn about is called the linear equation . | |
13:02 | That means it's an equation of a line . And | |
13:04 | we talked about that many , many , many hours | |
13:06 | long time ago in the equation of the line looks | |
13:08 | like this F of X is equal to a X | |
13:11 | plus B . Now A and B can be numbers | |
13:14 | , any number . So it could be two X | |
13:15 | plus three , three x minus five , you know | |
13:19 | , and so on . And you can think of | |
13:21 | this as the slope , whatever this is as a | |
13:22 | slope in front of this and this is the Y | |
13:24 | intercept . And the way that we find the roots | |
13:27 | of this guy is we solve the equation A , | |
13:31 | X plus B . Whatever it is equal zero , | |
13:33 | we know how to solve for X . We move | |
13:34 | be over , we divide by A . But in | |
13:37 | general what we give is we saw in the video | |
13:39 | , they're in general , the line could be horizontal | |
13:41 | with no crossing points . That's true . But in | |
13:43 | general it's going to have one crossing point . In | |
13:46 | this particular case , this is X . This is | |
13:48 | F of X . The crossing point is actually right | |
13:51 | here at this value of X right here . So | |
13:53 | this is one solution . And you can say it's | |
13:57 | like an X is equal to two or something . | |
13:58 | I'm just making this up . I'm just giving you | |
14:01 | an example . As we saw in the video , | |
14:03 | there's one crossing point . There's one solution because we | |
14:06 | expect one solution because the highest power of X is | |
14:08 | just equal to a one . Now , moving up | |
14:11 | the complexity a little bit , we have the famous | |
14:13 | quadratic quadratic equations which we're gonna be using and zooming | |
14:19 | in on almost entirely in this lesson . But the | |
14:22 | quadratic equations in general look like this . All of | |
14:25 | the quadratic equations have the form A X with the | |
14:28 | power of two plus some number times X plus some | |
14:32 | other number . Now A B and C can be | |
14:35 | any value Except for one exception . And that is | |
14:38 | that the power the value here a cannot be equal | |
14:42 | to zero . In other words , for it to | |
14:44 | be a quadratic , you have to have some , | |
14:45 | you can't have zero here , otherwise it would kill | |
14:47 | the quadratic term entirely . Right ? So in order | |
14:50 | to find the roots of the quadratic equation , how | |
14:53 | many crossing points you have , What you do is | |
14:55 | you take this quadratic equation , You have a X | |
14:57 | squared plus bx plus C . And you set it | |
15:01 | equal to zero , then you factor it . Or | |
15:03 | you could do completing the square . Like we've been | |
15:05 | doing , we're also going to learn about the or | |
15:07 | we will we will be doing actually completing the square | |
15:09 | to solve this guy will also be using the quadratic | |
15:13 | formula to find out the values of X that satisfy | |
15:15 | this equation . And then the main idea is what | |
15:18 | is the values of those that equation actually actually represent | |
15:22 | ? Well there's a couple of different cases and we | |
15:23 | saw some of them uh in the video , they're | |
15:27 | in the demo there . But let me just draw | |
15:29 | a couple of cases here , if your quadratic depending | |
15:32 | on the values of A . And B and C | |
15:34 | happens to look something like this and upside down Parabola | |
15:37 | , then there's two crossing points because you're setting the | |
15:40 | value of this equation equal to zero . So this | |
15:43 | is gonna be a solution and this is gonna be | |
15:45 | solution is gonna be two solutions , right ? So | |
15:47 | you're gonna have to solutions right ? And just to | |
15:51 | kind of make things up , you could say that | |
15:52 | X would be negative three and X could be positive | |
15:55 | , you know , two or something . I haven't | |
15:57 | exactly drawn it . But you see the idea you | |
15:59 | have a positive sum to positive negative value in this | |
16:02 | case you have to actual roots . What are some | |
16:05 | other cases that you can get with ? Quadratic ? | |
16:07 | So what does it actually mean ? Right . Well | |
16:09 | you can have a case like this . You could | |
16:10 | have of course a parabola that dips below the X | |
16:13 | axis like this . So you have two crossing points | |
16:16 | . Still you have a crossing point here in a | |
16:17 | crossing point here , you still have two solutions . | |
16:20 | You have two solutions and the values of these solutions | |
16:23 | are both positive . So it might be for instance | |
16:25 | , X is equal to one and X is equal | |
16:27 | to three . Those are two real solutions but there | |
16:30 | are other ideas and other kind of possibilities . Right | |
16:34 | ? You could have let's go over here and draw | |
16:37 | a couple of different possibilities . What if you had | |
16:40 | the parabola as we saw in the demo there floating | |
16:43 | kind of up above the X . X . What | |
16:44 | you see , there's no crossing points . There's no | |
16:47 | values where this equation actually equal zero . All of | |
16:50 | the values of the function are all above zero . | |
16:52 | So there are no crossing points . So there are | |
16:54 | no real roots . Right ? So no real roots | |
17:00 | . But what we're going to find is there's still | |
17:02 | two routes of the equation . They're just both complex | |
17:04 | , which involves imaginary numbers . Now , another example | |
17:08 | of when you would have no real roots is what | |
17:10 | if the equation kind of is down below , but | |
17:12 | going upside down . So you see all of the | |
17:14 | values are negative for this function . They do not | |
17:17 | ever cross the X axis . So again , there's | |
17:19 | no real roots here either , no real roots , | |
17:25 | right ? And then of course we have the one | |
17:27 | final example that we really need to talk about , | |
17:30 | which is the case that we kind of illustrated in | |
17:32 | the video here uh in the demo here just a | |
17:35 | minute ago . And that is the case of the | |
17:37 | double roots , right ? So what if you have | |
17:39 | the quadratic come down and just barely touch the access | |
17:42 | ? But only in one spot . It can be | |
17:44 | a positive oriented curve like there , or it could | |
17:46 | be in a negatively oriented curve where it only touches | |
17:49 | in one spot here . In either case there's still | |
17:52 | what we call to roots , but we call it | |
17:54 | a double route . And when I was learning this | |
17:58 | stuff the first time , the idea of a double | |
18:00 | root really confused me . Why is it a double | |
18:01 | root ? Well , it's because as I showed in | |
18:03 | the in the demo there , as you , as | |
18:06 | your quadratic goes closer and closer and closer to the | |
18:09 | access . The roots get closer and closer together . | |
18:12 | Eventually , when it's just touching the axis , there's | |
18:15 | still two routes but they're infinitely close together . They're | |
18:17 | basically on top of each other so they have the | |
18:19 | same exact value X . Is equal to one and | |
18:22 | then X is equal to one . The count twice | |
18:24 | . You still have to have two routes . So | |
18:26 | you still have two routes for this equation . Two | |
18:28 | routes for this equation , two routes for this equation | |
18:30 | . Two routes for this equation And even these equations | |
18:33 | have two routes but they're both complex and we'll learn | |
18:36 | how to find those when we , you know , | |
18:38 | continue learning how to solve the quadratic , so which | |
18:40 | is what we're gonna be doing mostly in this section | |
18:43 | . But this is an overview section and I want | |
18:45 | to talk about more than quadratic . We talked about | |
18:47 | this in the demo cubic equations . The general form | |
18:53 | of the cubic equation looks like this . Ffx is | |
18:56 | some number times X to the third power plus some | |
18:59 | number X squared plus some number times X plus some | |
19:04 | number . Right ? That's ffx that's a cubic equation | |
19:07 | . This number here needs to be something other than | |
19:10 | zero . But any of these other ones can be | |
19:12 | zero , like B , C . Or D could | |
19:14 | be zero because in other words it doesn't matter what | |
19:17 | the rest of this thing is . It's a cubic | |
19:19 | if the highest power is a . Three . So | |
19:21 | to figure out the roots of this guy , what | |
19:24 | you're going to be doing is you're going to solving | |
19:25 | this equation X to the third plus B . X | |
19:28 | squared plus C . X plus D . And you're | |
19:31 | gonna set that guy equal to zero . And we're | |
19:33 | going to learn as we get a little farther into | |
19:35 | our how to solve cubic equations . We're gonna learn | |
19:37 | how to factor things out and how to separate and | |
19:39 | divide things and we're gonna figure out the roots of | |
19:42 | this guy . But graphically what does it actually mean | |
19:44 | ? We saw in the demo uh in general what | |
19:48 | it means . But I want to make sure everybody | |
19:50 | understands now in general the cubic equations . I call | |
19:53 | them these snaky functions , they go up and then | |
19:55 | down the back up . So you see , there's | |
19:57 | three crossing points corresponding to the three solutions that we | |
20:01 | expect to find . So for instance , you might | |
20:02 | say that X is equal to like over here , | |
20:04 | -4 . You may see that this route is over | |
20:07 | here at -1 . And you might say that this | |
20:09 | route , is it positive to or something . So | |
20:11 | there's three real roots here . Three real roots for | |
20:17 | this equation . If this is what it would work | |
20:19 | to look like . But there are other possibilities which | |
20:22 | we saw over there when we did the demo there | |
20:25 | and there's infinite possibilities really . But you might have | |
20:28 | one that goes up like this and then snakes down | |
20:30 | but doesn't quite go below the axis again . So | |
20:33 | in this case you have only one real root , | |
20:35 | the other to what you have to have to make | |
20:37 | three or you're gonna end up being complex roots . | |
20:39 | So what you might say , for instance , is | |
20:42 | that your real root is gonna be access equal to | |
20:45 | like , let's say negative three , one real root | |
20:49 | and then the other two routes to more complex . | |
20:55 | Right ? So in other words , when we learn | |
20:57 | how to solve these things , we might find out | |
20:58 | that the real root is negative three and the complex | |
21:01 | roots might be one plus I that's complex number and | |
21:04 | then one minus I that's another complex number . For | |
21:07 | a total of three . Right ? And then one | |
21:09 | more quick case , since we're just trying to be | |
21:11 | , you know , explicit about everything , we might | |
21:14 | have something that looks like this . We might have | |
21:17 | the function go back to bend back below and then | |
21:20 | go up like this . It's a similar kind of | |
21:22 | situation . You have one real root here , right | |
21:26 | ? And you have two more routes that are complex | |
21:28 | , which we haven't learned how to solve yet , | |
21:29 | but they have to add up to equal three . | |
21:31 | So three real roots , three routes total only one | |
21:35 | of which is real . Three routes total only one | |
21:37 | of which is real . And we're only gonna do | |
21:39 | one more here . We're gonna talk a little bit | |
21:41 | about the court equations . The way you spell it | |
21:43 | is like this core core tech equations . Right ? | |
21:49 | And so the corn equation looks like this . It's | |
21:53 | 1/4 power A X to the fourth , B X | |
21:57 | to the third , C X squared D X E | |
22:02 | . So I have to use different letters . Right | |
22:04 | ? And if you want to solve this quarter the | |
22:06 | equation to figure out where the roots of the lie | |
22:08 | , then you just set this whole thing equals zero | |
22:09 | X to the fourth , BX to the 3rd , | |
22:13 | C x squared dx Plus the equals zero . Now | |
22:18 | we have no idea how to solve this . We | |
22:19 | haven't learned that yet . But there are methods and | |
22:21 | techniques you can learn to figure out the values of | |
22:23 | X . There's gonna be four values of X that | |
22:26 | are going to make this thing equals 04 solutions because | |
22:28 | it's 1/4 power . That's what we expect . So | |
22:31 | in general the cortex which we kind of learned , | |
22:33 | we looked at in the demo there in general is | |
22:37 | gonna look like this is going to be a w | |
22:39 | of some kind is gonna go down and up kind | |
22:41 | of like this right now . Of course it can | |
22:45 | be flipped upside down so it can look like an | |
22:47 | M . Right , but in general looks like a | |
22:49 | W . So in this case you have 1234 crossing | |
22:52 | points , so you have four real solutions like this | |
22:58 | . Uh And then of course we have other possibilities | |
23:01 | . Let's take a look at some of them really | |
23:03 | quickly . What if we had uh case which we | |
23:06 | briefly did examine something like this ? It goes down | |
23:09 | then up but not quite high enough and then it | |
23:11 | goes back up like this . We have only one | |
23:13 | to real solutions and too complex . We can't represent | |
23:19 | the complex solutions on this graph , but we know | |
23:21 | they have to exist because we know there has to | |
23:23 | be four total . And then we have the final | |
23:27 | case which I want to talk to you about and | |
23:28 | then we're basically almost to the end of all this | |
23:30 | stuff . The final case , if the cortex actually | |
23:33 | goes down and up and down and up and we | |
23:36 | never had any crossings at all , then we would | |
23:38 | have four complex solutions . Why have I gone through | |
23:44 | all of this stuff ? The truth is we're not | |
23:46 | gonna learn about court equations anytime soon . What we're | |
23:49 | really going to be doing is focusing on the quadratic | |
23:52 | equations . But oftentimes I know when I first learned | |
23:54 | this stuff , I didn't understand why we were doing | |
23:57 | it . Why do we care ? Well , it | |
23:58 | turns out that solving polynomial these are all different kinds | |
24:02 | of polynomial is really , really important . It has | |
24:04 | applications and calculus and physics and chemistry and engineering because | |
24:09 | a lot of times , even when you're solving really | |
24:11 | complex equations , which are called differential equations and calculus | |
24:14 | oftentimes it boils down to solving polynomial equations as part | |
24:19 | of the solution process . So we have to crawl | |
24:21 | before we walk right . But this entire set of | |
24:24 | lessons is going to be focusing on the quadratic equations | |
24:26 | . So I want to give you an intuitive feel | |
24:29 | because sometimes you're going to figure out that you do | |
24:31 | have the two solutions and sometimes you're gonna find out | |
24:34 | that you have two solutions that are real and two | |
24:36 | solutions in other cases that are complex . And I | |
24:39 | never really understood when I learned this the first time | |
24:42 | . When do you have the real solutions ? And | |
24:44 | when do you have the complex solutions ? This is | |
24:46 | what it boils down to lines . In general , | |
24:48 | crossing one location , you're gonna have one solution because | |
24:51 | the highest power of X is one . The quadratic | |
24:53 | are always going to have to solutions , but depending | |
24:56 | on where the graph is , those two solutions can | |
24:59 | be both real or both complex here we have two | |
25:01 | crossing points , so we have to real solutions here | |
25:04 | . We have two crossing points . We have to | |
25:05 | real solutions here . We don't have any crossing points | |
25:08 | at all . We still have to have two solutions | |
25:10 | , but they're both complex . When you actually solve | |
25:13 | those polynomial that would give rise to these graphs , | |
25:16 | you'll find you're gonna get with quadratic formula which will | |
25:18 | learn soon , you're gonna find that you get imaginary | |
25:21 | or complex solutions . And then we have the cases | |
25:24 | where the graph just kisses the axis and we talked | |
25:26 | about why these are double routes . They count as | |
25:28 | two routes as well for a total of two . | |
25:30 | Then we have the cubic equations which look like these | |
25:32 | snaky functions of course . Remember you can mirror image | |
25:35 | , reverse them so that they go up and then | |
25:37 | kind of down like this in general , they're crossing | |
25:39 | three locations . Three routes for a power of three | |
25:43 | . In this case it doesn't look like there's three | |
25:44 | routes really , but there is because there's one real | |
25:47 | root and then we have the two imaginary roots are | |
25:49 | complex routes that would make for a total of three | |
25:51 | here . In this case we have one real root | |
25:53 | and then the other two would be imaginary or complex | |
25:56 | roots and then we have the cortex where in generally | |
25:59 | we have four crossing points . But depending on where | |
26:01 | the graph is , two of them might be real | |
26:03 | and two of them might be complex . So that's | |
26:06 | why when you're solving these equations and folks are gonna | |
26:08 | be focusing on the quadratic . Sometimes you'll get real | |
26:11 | , sometimes you get complex . What it means is | |
26:13 | if you were to graph those functions that tells you | |
26:16 | what the graph really looks like , it tells you | |
26:17 | where the crossing points are , how many of them | |
26:20 | are there . And that's basically what it boils down | |
26:23 | to . Um and that's all I want to really | |
26:25 | get into now . So what we're gonna do is | |
26:26 | we're gonna be learning how to solve quadratic equations with | |
26:29 | new methods that you haven't learned before . They're among | |
26:32 | the most important topics and types of equations . We're | |
26:36 | gonna learn how to solve that have applications and everything | |
26:38 | , so we're gonna spend a lot of time on | |
26:40 | it . So follow me through the next batch of | |
26:41 | lessons and we'll be covering the solution methods for quadratic | |
26:44 | equations in great detail . |
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