01 - Visualize Roots of Equations - Linear, Quadratic, Cubic, Quartic Solutions - Free Educational videos for Students in K-12

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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