Algebra - Completing the Square - Solving Quadratic Equations - Free Educational videos for Students in K-12

Algebra - Completing the Square - Solving Quadratic Equations - By tecmath

Transcript


00:01	Good day and welcome to the Tech Math channel .
00:03	What we're gonna be having a look at in this
00:04	video is we're going to continue looking at some basic
00:07	algebra okay . In a few little ways of expanding
00:11	and facts arising and all these special little things that
00:14	we like you to do in algebra and that sort
00:16	of deal . And it's a really good thing to
00:18	get worked out . So when you get on to
00:20	more complex , where if you have to do this
00:22	, it's not such a hard thing . So let
00:25	me go through a couple of rules of what we're
00:28	having to look at here . So first off ,
00:31	we're gonna be having a look at the difference of
00:32	two squares rule . Also , we're going to be
00:35	having a look at perfect squares and we're gonna be
00:38	looking at these . Okay , so first off ,
00:41	I want to actually have a bit of a look
00:43	at what is meant by this idea . First off
00:46	, what we're going to be having a look at
00:47	roughly is where we've been having a look at these
00:49	sort of questions . Okay , well we're trying to
00:52	expand these sorts of problems . Okay . And you
01:00	might have , if you've been looking at these earlier
01:02	videos , you might have realized that the answer to
01:04	this one , you know , we've got X times
01:07	X , which is X , where we're gonna do
01:09	X plus three and X times two . So all
01:14	together , that's plus five X . And then we
01:17	have three times two , which is 15 . Okay
01:21	, so we can expand these out . We're going
01:23	to be having a look at a little tweak on
01:25	these and how you can get some slight variations to
01:27	this . Okay , now I want to start out
01:31	actually having a bit of a look at this idea
01:34	of a perfect square . So perfect squares , um
01:38	a perfect square is a number of pro numeral and
01:40	expression , which is a square of another number .
01:43	Pro natural expression . So what do I mean by
01:44	this ? Well , if you can imagine one or
01:48	four or nine , these are numbers , which basically
01:54	the square of another number . Okay , so you
01:56	can imagine that one squared is one or two squared
02:00	is four . Three squared Is nine . OK .
02:04	You might actually even imagine this as a square .
02:07	You might even think of a length and this being
02:12	an area . Okay , so if we have a
02:15	length and we had a square , Okay . Um
02:19	so say we had to say something like this is
02:21	hopefully it comes out of the square , something like
02:23	this , a bit of a rectangle . Looking at
02:25	square , this one . Okay . But what we
02:27	would have is we could have a side length of
02:29	to the side like that too . And we got
02:31	this area for now , we can use this idea
02:34	also in algebra because say we have , say something
02:39	like a perfect square would be called say something like
02:42	imagine if you had X squared ? Well this is
02:48	the same as having the side length . If I
02:50	could draw this square once again of having the side
02:53	length of having X and X . Okay . Or
02:57	we could have nine X squared . Okay , where
03:02	what this is is we had three X by three
03:04	X . We could even Okay , so I'll put
03:07	this down here . We could even go this next
03:08	step where we have this sort of thing . X
03:11	plus one , swear . Okay , and if you
03:15	can imagine this , X plus one square , this
03:19	is like having X plus one times X plus one
03:23	. So this is where we have X plus one
03:26	is our side length . Okay , so these are
03:28	all ideas of having this perfect square , and we
03:31	have been looking at how to play around our soul
03:34	. There's a couple little shortcuts with this . Okay
03:36	, this is the perfect square shortcuts are fairly uh
03:39	similar shortcut to the one , we were just having
03:42	a look out there . Okay , but I first
03:44	off before we do that , I want to get
03:45	into this idea of this , first look at this
03:48	thing , of the difference between two squares . Um
03:52	Yeah . Mhm . So say we actually considered expanding
03:56	this particular expression ? Yeah , so I gave you
04:00	this one here and it was a take a big
04:04	and in brackets , I gave you a plus B
04:10	, and I said , okay , I want to
04:11	expand this here , and if you would expand this
04:14	, you would get the following . Okay , and
04:16	this is using that same sort of idea . You're
04:18	gonna start seeing I have a bit of a way
04:20	that I do these but we have I'm gonna I'm
04:25	gonna actually do this in the long way . We're
04:26	going to start multiplying this one by this one .
04:29	So we end up with a squared , we've got
04:32	this number I college be so plus a babe .
04:36	We have this number for this number minus B times
04:41	a c minus abe and we have this number minus
04:45	B times minor times positively , which is minus B
04:48	squared . Okay . And what you're going to realize
04:53	possibly is that we have positive ab and negative ab
04:56	. These guys cancel each other out . So what
04:58	we're left with is this I squared take away B
05:05	squared and I put it up there because what we
05:08	actually got there is a handy little rule we can
05:10	actually start to use and I'll show you how this
05:12	actually goes . Okay . So I'll get rid of
05:16	this extraneous stuff here and just see this particular rule
05:20	that we're left with here to say you were asked
05:23	to actually uh this this just before I go on
05:28	, what you notice is here , we have a
05:30	difference between two squares . This is called a difference
05:32	of two squares rule . Okay . We have one
05:34	square here and the two squares there . So it's
05:36	called a difference . Because the difference . You work
05:38	out by subtracting . Now , We can use this
05:41	rule to solve a bunch of different uh expands and
05:45	different types of our expressions . I'll give you an
05:48	example here , say you can do it fairly immediately
05:51	. So you had to say something like 9 uh
05:55	take away X . And we also had nine plus
05:59	X . Okay , these both and brackets apparently sees
06:04	there . Now , how would you go about doing
06:06	this ? What you're going to realize is we can
06:10	actually follow this particular rule along . We don't have
06:12	to go nine times nine is 81 . 9 times
06:16	x is nine x minus nine X . And then
06:19	we get minus X squared because we can actually follow
06:21	this particular rule . This is one of these certain
06:24	types of ones where we actually have the positive and
06:26	the negative here and the positive there . Okay .
06:29	So if you see one of these expressions , what
06:30	you can do straight away is this You can look
06:33	at a here and square it . So nine times
06:35	9 His i . d . one and we're going
06:39	to take away this time this X squared . There's
06:44	your answer . Okay so this is a fairly immediate
06:47	type thing you can do using this rule . Hopefully
06:50	you get that will give you another example here .
06:52	So we can say something like well what about we
06:55	do X plus three X -3 . That doesn't matter
07:03	whether this is positive three or this is a negative
07:05	three then this is positive three . But as long
07:07	as these ones we got the same letters here ,
07:11	we've got the same numbers but one positive and one
07:13	negative . Okay , so we can immediately and so
07:16	this one and we're gonna X squared And here we're
07:20	gonna end up with -9 . Okay . Past three
07:23	times -3 . Okay , we can get this immediately
07:26	and this is where this difference of two squares rule
07:28	is really , really handy . Okay , um what
07:31	about we do a bit of a harder one ?
07:33	What about I do uh eight take away five A
07:39	. And you can probably guess what I'm going to
07:40	have on the other side here is going to be
07:42	eight plus five . So if we were to expand
07:46	this out straight away eight times 8 64 and we're
07:50	gonna be taking away five times five first , we
07:53	do that coefficient first , and then I squared okay
07:58	, we can do this immediately . So how did
08:01	you go with those ? So that's a difference of
08:04	two squares rule . Now , I want to so
08:06	that's expanding using that particular difference of two squares rule
08:08	. And so it's a really , really short cut
08:10	way of doing things , which I think , you
08:12	know , you want to be able to do fairly
08:13	quickly . So that's one of the shortcuts you can
08:16	use we also have this idea of expanding using the
08:20	perfect square rule and oh , just go through this
08:24	really , really quickly . But I just I'll show
08:27	you how this works , but I'll show you how
08:29	I prefer to do it . So , say you
08:31	give an expression like this , This is the expanding
08:34	using the square perfect square rule . So we have
08:36	A plus B . And this time we're squaring it
08:40	. Okay , so we've got this perfect square here
08:44	and we want to expand this which is equal to
08:47	A plus B . I I plus B . Okay
08:54	, now we can go through this right now and
08:57	we can do this type of thing . Where what
09:00	we did is we would go eight times eight is
09:03	a squared , A times B . Okay , A
09:07	times B is a B , B times A is
09:11	a B . And B times B is B squared
09:16	. And we're gonna end up with a squared plus
09:20	to a B plus B squared . And this is
09:24	the rule which is known as the perfect square rule
09:27	. Okay , Where you end up with particularly these
09:31	ones here . Okay , so this is equal to
09:37	you see ? Yeah . Ok , So look ,
09:43	that's all well and good . And I'll show you
09:45	an example of where you could use this , right
09:47	? You could easily use this where first off you
09:49	did . But I want to show you how a
09:51	little bit how I'd actually otherwise tackle because I think
09:53	this is , you know , this is all well
09:55	and good to realize this perfect square , but I
09:57	have a different way . I prefer to take all
09:59	these . I say you've got to say something like
10:04	What about we do four plus x and we square
10:10	this . Well this is going to be equal to
10:12	a squared , which is four squared , which is
10:15	16 plus two . Ab Okay , this is A
10:19	and this is be so four x times two is
10:22	eight X plus B squared plus X squared . So
10:28	you can immediately get these out years in this particular
10:31	rule and you can see how far that can be
10:33	right ? But I'll tell you how I actually also
10:36	, and especially if you have the expression like this
10:39	, I think that's not a bad method of doing
10:41	it . But for me , also , if I'm
10:43	actually giving a little bit differently , if I'm actually
10:45	was given it like this uh four plus X .
10:52	Four plus X , another way you can do this
10:54	just to confirm , obviously you can expand this normal
10:57	way . I also look at it go I square
11:00	these two numbers first four times four or 16 .
11:03	That's the first part of our expression . They're not
11:05	gonna have four X . And four X . 1
11:09	to put them together . Okay , Four X .
11:11	And four X . Eight X . And then I
11:13	just stand up with the X times X . At
11:14	the end of this is this idea of this rainbow
11:17	multiplication , which I've looked at in other videos .
11:20	Okay , so one last example of this , What
11:26	about we do this type of thing where um I'll
11:30	make a bit of a more difficult one . What
11:32	about I actually do this . Okay . Yeah .
11:36	Actually , what about we just give you , give
11:38	you a way to get over it . Uh What
11:39	about two X plus ? Uh four And I want
11:46	to square this . So this straight away is a
11:50	square which is four X squared . Okay . To
11:53	abe , Which is this times this ? So eight
11:56	X two x times four is eight X times two
12:00	is 16 X plus and then plus B square ,
12:05	which is plus 16 . Now , I just want
12:10	to actually I just want to go through one last
12:13	one of these Because I think we haven't actually gone
12:16	through one of these where we have a negative here
12:18	. So say we have uh this er and we
12:21	have -5 squared equals . All right , So let's
12:28	just do this really quickly . Two x times two
12:31	x is for X squared . Mine too , Times
12:35	minus five is minus 10 X . Two X times
12:38	minus five is minus 10 X . Times two is
12:41	minus 20 X . And then -5 times -5 is
12:46	plus 25 . Okay , so how did you go
12:51	with those ? They're just handy little tricks to know
12:54	, especially a bit later on when we're going to
12:55	be trying to factories this and we're trying to go
12:57	from this type of expression and send them back this
13:00	way . But we have some little tricks for doing
13:01	this as well . Okay , so hopefully you feel
13:04	okay with these . Um it's just an extra little
13:07	step on being able to work with work with those
13:10	. So next we're going to start having a look
13:12	at Factory Ization , which is basically turning this type
13:15	of expression back into this . It's a little bit
13:17	harder , but it's not that bad at all .
13:19	So I hope to see you then bought .

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