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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