Mhm.


You take a prime number. So prime number, you know, 2357, stuff like that, right?


mhm


So you just say, OK, my first smallest brand number is 2. Good.


Then I get a 3 year, correct?


Mhm


What's my next plan number? 3


I got a 1 here and I stop.


So now you know your two factors of these 3 and 2.


You just sort of walk signs now.


Do what? could you


All right, so why are we doing uh


that


I mean


Why are you doing that? So you look at this number 6 here, um.


you know, this number 6 here.


in the equation, right


uh-huh


So that's the reason we're doing 6. Is that the question? Why did we pick 6?


Yeah, but like, why are we finding then the prime number of 6.


or like the prime


You, you could use anything but Prime makes it easier to search for them.


I mean, if you, if you have a better way of doing it, we can do 6 multiplied by 1 S 6, but then it won't give you the factor of, you know, so let's do that. Let me try, try doing the other one.


Oh, so you're doing, like you have like for me when I look, what I'm looking at is like it's 2 and then basically that corresponds with 6.


I that


Yeah, yeah, you can think of it that way, but you know, it, it works for this problem. It may not work everywhere.


Yeah, like you don't, you have to change uh 6 or


Yeah


Mhm.


So it, it may get uh complicated. So let's take this for example, the problem that we had in hand.


So do you, you understand prime factors, right? Is this very clear, uh, the one here in this


uh


yeah, but like you could also use like for 6 you could do 1


well no


one is in prime, right


OK.


Yeah, you could do 1 and 6, for example, if I draw this one more time. Let's do this, uh.


I could do it this way, I call it.


Like 2 and 3, right


Yeah, yeah. Yeah, it's 2 and 3, right? That's what we're doing 2 and 3.


Mhm


So then the two in like the 3 that's on the bottom goes basically to the 6 and it's connected to it.


or is it connected just to the three basically.


Like,


Hm.


No, no, so this will be a factorization. What factorization means is, uh, so let me explain what is done here. So suppose if I have 27, for example, right? So does that, is it duzzled by 2? No, right?


So 3


It's 3. So you do 3


927, correct?


Three multivitamins 27, correct?


Yeah, but why do you, so the way you're doing that is like you're multiplying


you're putting the number that's multiplied by


below the number you're getting.


Yeah


and then you're finding the factors of that number you multiply by.


Then I, yes, correct


That's a method of finding factors.


OK


So you could do this like 3 and then 3 is 9, correct? These two multiply gives you 9.


and then 3, multiplied by 1.


uses that 3


So the factors of 27 are 3 by 3, by 3.


Oh, so like the smallest, so that's the smallest possible thing.


Correct, the smallest possible you start and you start with something that's smallest but prime.


OK. But then couldn't, and then this could also work though if you


just


like with any other number, right?


Yeah, it can work any number that has factors. So sometimes, for example, if this is, this is 29 for some reason. It won't work, right?


Yeah, but there won't be any factors and it'll just be 29.


Exactly, that's fine, correct. Exactly.


And then this goes into the problem so that, I mean, I know it's like to find the factors, but like,


Yeah, the idea is you're able to find the factors for 6 here quickly, but you know, this was easy, but if it's a complicated one.


oh, wait, wait, wait. So, the reason we found the factors was just so that we know what numbers we can multiaby to get the 6.


Correct, exactly. X gives you 6, and then you use the signs here, minus and plus to get us this number, which is an addition, right? So if I use both as positive, 3 + 2 is 5.


So that doesn't work


If it is negative, then it has to be -3 and +2.


Correct?


Oh I did. So, we, why are we doing the addition and it's


Because if you add


suppose if we add 2 and 3.


what do we get here?


you would get 5, but like.


Right, but we don't want the 5, right?


What you want is plus one


Oh, wait, wait.


You want the one here, right


01, like you want a one in front of, like, do you, like, you want a 1 for B, right?


Correct. You want to be equal to 1, exactly.


Because that matches the equation.


Exactly


OK.


So you have to think in such a way that how do I take this, now I found a factor that multiplies to 6, that is 2 and 3. We know that. So now the next step is, after we have checked for this, you have to check for this one. OK, now I want something that adds to one. So what can I add to one? If, if it is both positive, it adds to 5.


If it is


Should we do mm


yeah, but


I mean, you would do a 3 minus 2.


These are your three options, right?


Yeah, OK, so you just figure out which one matches to get what I want.


Which one works for you. Correct, correct. So this one works for you basically at this point here. + 3 minus 2.


OK, some something similar than in the past where I just find the factors


like if you scroll down


um, oh wait, I don't actually have it on this one, but like I would add the different values together and at least on like later assignments I would do like 1 minus 36, 1 + 36, and then 36 minus 1.


to get the 3 versions


Yeah, yeah, that works. Yeah, those are the simpler versions. I think now it's getting complicated.


So coming back to this one, let me just delete what I've written here.


OK, let me just uh get fixed this. All right, there you go. So what now, now that we understand the the things on the right. Let me just, uh, maybe add one more color on the thing on the right.


So what you're essentially doing is you're also solving for the one which is here. So you see this?


You understand there is a one ahead of X, right?


Like there's a one ahead of you


Uh, yeah, that's A, no?


Correct? Yeah, that's A. So A is equal to 1 and then C is equal to 6, and you always turn to the factors of a multiplied by C.


Wait, wait, uh, C would be, why is it 6 or 6. It's just like


is this the standard like equation just minus 6 or


It could, yeah, it is -6. Let's see equal to -6. You're right.


like minus, I mean


C is -6, C is equal.


uh or like does it just not matter that much?


No, no, let's, let's stick to that. That's fine.


OK.


All right. Uh, my, so similarly, you need to find out what's here. A is equal to


3, correct


All right. Also, is it like, is it like A times C equals B or?


Yeah, so you need to do the, what you did the factor for 6 years is A times C.


So you need to do A times C is 12 times 3.


So we need factors of 36.


No, wait. So A times C is B, right?


No, no, not 8 times is


not going to be


OK, so


that it's just saying that earlier. OK. Well, um


yeah,


OK


Hidden is not going to be, so did I say that, uh, yeah, no, no, I, I didn't mean to say that. 8 times he is not equal to B. What, what, what is happening is the factors have to add up to give you the bean.


like you did here, right


Sure.


factors to add up to you be. OK, so then, yeah, like 3 minus 2.


Got it, got


And then


OK, yeah.


So if you take 36 and apply the same method we did, so the smallest prime number is 2, correct?


And we're doing 36 because


oh yeah, so we're just multiplying A times c to get 36.


Correct.


Uh


Here also multiplied it by uh A C, but the CA is 1, so it didn't matter.


So you do the smallest factor 2 and if you do 2, you get 18, correct?


Uh, OK, so like,


so the verb, no, no, no, because they have that on the work too.


Ben, I have it on like the right side. It's


it's like Ax C equals 3 times 12. He was 36.


And then it's 2 times 18 and the sum of that is 20. So the sum, the sum of the factory.


where


Yes, so you don't need to break it down any for correct, correct. You don't need to break it down any further now because, you know, it to and 18 using the 20.


I could have a second, uh


Sure.


I see that do my. Sorry.


I'm doing tutoring.


Let me know when you're back


Hello


He


Yeah, I'll be back


Yeah


OK. So I've written down the equation based on the two factors that he shared, the 18 and the 2.


Mm


You see this


OK, so


you


The 18 + 2 is 20, correct?


Wait, wait, so we did a 3 X squad.


or 3 times y + 18 x + 2 X.


+ 12


OK


So what we did is we took the 2 and the 18 that uses 36, right? So 18 + 2 is 20.


Mhm


Wait, so what's


And we know that uh mhm.


that the slat eagle's beak.


18 + 2 equals B then.


Correct, correct


So.


Yeah, so the, what was it again?


was


the fact, yeah, those are like 2 and 18 are the factors of


A times C that are equal to B.


right? Yeah


Correct. That's what we got here.


It could


could I like just like write a note like above.


So


Sure, this, I think the link is gonna get forwarded to you on email.


Oh, OK, OK.


But yeah, if you wanna write notes and it'll be, it'll be on that link. Nothing's gonna go.


Right.


So, 36 U 18 and 2, and now you break it down, right? So 3, you can, as common, X is common, the 3 eggs.


and then what do you put in the bracket is X.


Are you with me so far


Um, yeah, so that would be 3 x + 18 x.


Correct.


And then where's the 2 X? You're just adding it after?


No, and then you do the 2 common


because you want the X + 6 to repeat itself now on this side, right?


So you take the 2 out and then the X + 6 will repeat itself.


OK


11, right?


Yeah, so it's equal now


Now it's equal


But uh, like I have it like for the other one, like where I normally do it, it's A times the parentheses the parentheses like there's no like in between number.


Uh


yeah, yeah, that'll, that'll happen now. So the next step is


I'm just breaking it down for you. Um, so between in this equation what is common now? The X + 6 is common, right?


Mhm


So when you pull extra seats out from here, you get 3 X's left, correct?


Oh, it's, so


um


That's


I just clipped it


Yes, you just inverted it, right?


Correct, correct


OK.


Yeah


Just flipped it over and you take the X + 6 out here.


and you get the 2 out here


Um


uh


So 3 X X + 6 multiplied by 3 X, which is what you have here.


Yeah, so you're doing, yeah, but then you're doing also


And then you have the two muddy bad bags, the X + 6, which is the second part here.


OK


Yeah, like the way they would make us do that that is like you do.


like right something quickly.


Correct


Uh


zoom in. I know you can't zoom in.


How do you, yeah.


And it is it


Uh, yeah.


Do you wanna erase the line or?


Yeah, yeah


had you like erased the whole line?


So you just click on it, so you have, uh, you have a bot, right? Bot.


Yeah.


Go to the right of that you have this dotted square.


You click that and you select.


like I did right now


OK


OK, so XM 6 times.


move


What, how is it right like that?


It's not good


What is that? Sorry


Yeah, it just doesn't like


No, so here what you're doing is that this whole equation is equal to 0, so X has two values essentially.


So the first value is -6.


and the other value X can have is


-2 x 3


So if you put 6 in here plus 6 right, minus 6 + 6 is 0, right?


OK, wait, wait. So


look, yeah, I'm pretty sure like they are


like they are equivalent, but, uh,


Chex, yeah


Yeah.


Yeah, OK. Because I don't know what you, how you did the X's negative.


because


yeah.


60, OK, OK. So you're like figuring out


basically if


like what X would be if X + 6 equals 0, right?


Exactly. So when you're multiplying either any one of the numbers is 0, the multiplication is 0, right? So the first assumption is that X + 6 is 0.


then the whole thing will be 0


So that's when you get X is equal to -6. And the second assumption is 3 x + 2 equal to 0.


So in that case, X will be minus 2 by 3.


-2 divided by 3


Yeah


or 2/3


And then that's it, huh


That's it. So you have two values when the whole equation will be equal to zero and you solved it.


Oh my God. That's so long.


Easy peasy, you


So long and so short


OK


Yeah. The key is to figure out the factors, so you know you do more practice and then you figure out the factors more quickly, um.


It's just like


with the work they gave us like


what uh


I think is like gluing out.


yeah.


But it's like on a full gray screen. What do I do?


Oh, never mind. OK.


Um, but like, did you get what was on the page?


Yeah, so in that they are um


doing the same things but trying to figure out the factors. That's what they're doing.


OK, so, so can you give, so give me like, could you give me like a quick thing on just what all the steps are?


Like quickly, so like first


Like what they're doing


actually, no, I'll try to do that and then you just like, could you like tell me like if I have something wrong?


Oh, for which one, sorry


The second one


No, no, no, like, so.


no, I guess both, but


I think we just did the first one.


the first, yeah.


Mm


OK. Um


for the second one


Are you asking that you want to do the second one and I should take a look, is that what you're saying?


I don't know, like I just like list the steps quickly cause the problem is just necessarily like the actual work but more so just the steps cause the actual work itself isn't that hard.


Oh, OK, OK, I see your point.


Uh,


OK


yeah, I think that steps is this, um, I need to dial out a bit, so that's why I'm gonna type it out to you and see if you understand it. So step one is uh do uh A.C.


Correct


Yeah


Well


I thought it was like actually um you just convert it into the


parentheses parentheses form first.


or do you need to do 8 times C first and then do the.


No, you won't get the pans until you find the factors.


OK, OK


So


You need to do the A or C first, so you get the 36, right in the example.


Then you find factors of


8 or 6


So for that one, it would be like


AC


It's like lagging know for


right


Step 1;  a.c
step2:  factors of a.c using prime numbers


Let me see. Yeah, OK. So then


Yeah, I'm writing practice for here it is. Step 1 is A or C. See, you see that? I just wrote it.


they were


You can see on the chart, you can see it


Oh, OK, OK.


The step one is ARC. Step 2 is factors of ARC using prime numbers.


and then step 3.


is uh A


So a little bit some like a notes thing quickly.


So


Yeah, B is equal to


addition


of factors of ARC.


step3: b = addition of factors of a.c


And that's it.


so 8 times C and then the factors.


of the


times C


and then


sum should equal B


right


Wait


Be because of some factors of ARC and that's it.


some of the factors of A, let's see.


Yup


equals B


you would just find then uh


but after that you would get like, for example, the other problem it would be 2 and 18 or 18 and 2.


Uh.


Yeah, that is like I mean it's, it's be straightforward.


So then I, but then wouldn't it be a problem though, or


Sorry


wouldn't there be a problem though? Cause right now, cause it's like you're doing 18 X + 2 x or 2 x + 18 X.


Yeah, it's the same


So it's fine then, no?


Yeah. So, yeah, yeah, that's your work. So I'm gonna share a link here. You can refer this as an example. I never dial out, sorry.


Do you see the link?


Uh, well.


Yeah


Uh, I can put it here too on the chat. Do you see it now?


I put it on the board too


Oh yeah


Wait, can I like


OK, yeah, I can


You can copy on the board, yeah.


Can I open the thing up


Yeah.


Uh, you learn to copy paste it


No, like I already like copy and pasted it.


Oh, OK, yeah, you can open it up. So it, it gives an explanation of how to solve it also.


OK.


Yeah, but why aren't they making us do the quadratic.


Like it seems weird, no? Or


What do you mean? Why they're doing the


like why don't, why don't they just let us do the quadratic formula, so just to get uh both versions of X


I didn't get the question, sorry


Wait, like, cause in this, they're making us do like all of that stuff.


But like we can also use the quadratic formula to get X on


So you're saying that can we use the other method like the formula that you already have.


Yeah


Yeah, yeah, you can use that too. Uh, that's the other way. So if you see.


I think the equation is


Like I was just like saying that because like it seemed like it could be simpler to do that.


Like the math might not be as simple, but like I just feel like the process is like


it's less


like complicated I guess


x = [-b ± √(b^2 - 4ac)]/2a


Yeah, yeah, I mean, you can use, if you want to use the roots, that's fine because, but you need to figure out then the like you're talking about this one, right? I just typed it on the screen.


Do you see it?


Mhm. Yeah


Is that the one you're referring to


Uh, see, yeah.


Well


Oxycod minus B, that one.


yeah


Yeah, yeah, yeah, you can use that too. I mean, uh, if that's easier for you, that's a quick way of solving as well.


But I, I think they're trying to show you alternate methods of the same thing because if you just put a formula, you don't know what you're really doing, right? But if you, if you do it this way, that you know you know that your factorizing.


the quarter day equation


Oh, no, no, I don't, I don't forget it either way.


I'm just like trying to figure out


how to like remember how to do this, but like, once I do it more, I, I think I will try to, or like I will start understanding it though.


It's like I've only done it once.


like


Yeah, yeah, I mean, once you start doing factorization, I think you will continue doing this and you will not go back to that uh formula.


becomes more fun.


OK


because this becomes more fun, I think, at least in my mind, but yeah, give it a shot and see if it works for you, because putting the formula is more mechanical, you know, you're not applying your brand.


Yeah


But if you like that, that's the way to go to.


OK, so


Because sometimes when you go to trigonometry, you will start applying this formulas. You won't be applying, uh, you know, methodology.


So yeah.


So you get the similar factors of a times c which


uh in that, in the first problem would be 20, and then


hm


Yeah, in the second one, if you wanna talk through it, that's fine too. I can, I think I have some time now, um.


wait


cause you do like uh 3


X and then X or entheses X + 6 + 2


Mhm.


times X plus or minus + -6 and then you convert that into


X + 6 times 3 X plus 2.


But uh where does the the 18 come in? Like


18 and 2 in the addition that makes it 20.


yeah, oh, OK, OK. But


why is it actually matter that we do that then because couldn't we just


Otherwise, you won't be


oh wait wait, cause you're


we


Yeah


Yeah, yeah, OK, cause you're converting the 3 X


2 plus 18x, and then you're converting that into


yeah


yeah, OK, OK, I see what you're doing, you know, cause the plus is still in the parentheses. So you're basically just simplifying X. You're basically just doing um


I'm factor that factor.


Right.


3, yeah, yeah, you're doing, yeah, you're factorizing it. You're just doing 3 X X plus 6, which is the same thing as 3 X plus H X. OK.


right


clear


So the same way you should think about this A.CS 40, I think you started writing that. So A.C is -10 and 4 is 40.


And, and


I'm gonna dig a screenshot of the, of the beautiful work.


OK


Hm


OK. And that's X


And then


Mhm


one last thing before we go. So for question 2, I just want to do that really quickly. Like, so 8, some of the factors for it


A times C.


So you would do AC that would be -40, and then the factors


Mm


that would be equal to actually question now, would they be equal to 3 or -3, 3, right?


I mean, I mean just, yeah, I mean, forget the negative, positive for now. Uh, you, you can just think about the signs later. Just think about how do you break down 40


so that you can get a 3.


Oh, OK


But like consider them like when you're looking through each of them to get to the actual being.


Yeah, at that time you'll make signs. You're right, you're right.


OK, so


Yeah


1 in 40 will never work. 220


that work


Uh, 4:10


no


You know, that wouldn't work cause


OK


Yeah, think, think prime numbers. It's easier to think prime numbers.


Oh, yeah, from phone numbers.


uh


So 2 then work, 3 didn't work. What about 5?


50 yeah, the first prime number.


Mhm


Uh, OK.


Yup.


That's


So that would be 5 and 8. Yeah, I'm going to do some of the prime factors. So are they just normally like the right ones?


go with


more likely or not because you don't want it to uh


uh, yeah, no more than uh more, more likely than not. Let me say it that way. If not, you know, if you there may be a case where we have to multiply a couple of those prime factors, but it's always the cleanest way.


OK, fine. I'm sick writing this and basically we find the


prime factors of B, which is equal to AC.


So you got 55 and 8, right? So does 5 and 8 work?


Uh, 5 and 8, yeah, it works because you're doing 5 minus 80 equals -3.


There you go


OK, so then the equation would be


you would basically do uh I mean they write it down, it's gonna be like annoying.


Yeah, it's gonna be -8 X +.


Yeah.


OK.


really, I gotta convert it. You're too too smart for me.


No, no, that's fine. It will take time.


know.


Corax square +


5 X minus 8 X.


minus 10, no? And then cause that that'd be the same thing.


And then you basically just convert that to a simpler form. So,


would do for X


We, yeah, 4 X X plus


Could you do that


Core, sorry, what did, uh.


A 4 x times entheses X + 1.25.


why do you do that


because, uh, 4 x times X would be 4 x squad, which is


A x1 and then 4 x 1.25 would just be 5 X.


No, no, no, no, no, no. I think you're missing something up. Why don't you use the 8?


and then


hm


No, no, use the 8. Why are you using the 5?


Uh


Oh, wait, so you can use, wait, can you just use either one then, just whichever makes more sense.


Yeah, yeah, you need to use the one that, that gives you the clean factor.


So the negative 8 X + 5 X.


Yup


minus 10. OK. So then 4 x parentheses X + 42.


Yup.


or 2


cause it's minus 8X, right?


Mhm


or


plus or times


oh


So you already got X minus 2, right? So the second factor should also give you next minute too.


Yeah.


So the X minus 2 should repeat itself. Otherwise, you cannot take it as common on your next step, right?


Well, I just disconnected. Uh, you hear me?


Yeah, I can hear you


And my fingers stillleted, but I remember so much, uh, so it, it was 4 X X minus 2, right?


4 square minus 2. OK, and then so above it


was


the rest of the stuff was 5 x minus 10. OK.


So 4


No, no, so hold on, 5 x minus 10 will, you can take the 5 common, right? And then that would be.


X + 2


Oh, because I was just writing down what was left from before.


Oh yeah, yeah, OK, good, good


Sure, sure


like the like the whole thing was 4 x squared minus 8 X + 5 x minus 10.


Got it


And then I got that down to 4 x x minus 2, and then just parenthe again, no? Or


Uh no, just + 5.


oh wait, yeah, so then plus


oh OK


plus 5 x minus 2 again because that'd be 5 minus 10.


and a plus 5 parenthesis X minus 2.


Yeah.


Yeah, yeah, yeah, that's, yeah, that's what I meant.


Yeah, yeah


OK, and then


hm.


what did you do then? You would you just as so.


Now I minus 2 is common, right? You have 2 X minus 2s, so you take that one out and then put the rest in the parenthesis.


So you remove the common factors?


Yeah, take the compactor out.


OK. OK. It's equal to AC and then


right


Uh, just give me a second writing this down in my precious notes.


Yeah


No, good, good. That's, that's good. You're diligent.


factors.


What


Uh


OK. 3 X is like the equivalent sign.


it's like a version.


that you might see


like.


like seeing streets.


So.


That's what I said. You just gave an example.


in both circumstance.


frontage than right.


In fact they get So you have to avoid decimals.


right


What's that


So you have to avoid decimals, right?


Yeah, so just


Yeah, I mean, mostly inquire, you'll get clean factors. You'll never get fractions. Are you, yeah.


so what about if we like avoid dusts if you can, but if you can't, you just end up with decimals, no?


Or is it like you never do?


Uh, so, no, no, no, no. Let's, let's just think about the other way. The idea is to find that common factor like the X minus 2 that you found here, right?


So


No, I was like talking about um like I don't know if you were like talking about just action, but


like for the 4 x 2 minus 8 x + 5 x, like you have to avoid doing 5x minus 8 X because you you end up with the decimal like you end up with like


yeah, yeah, I, I see, I understand your question. What I'm trying to say is that instead of thinking about it that way, that you need to avoid decimals. The way to think about it is


how do I find a factor that'll be common for both the 1st 2 and the 2nd 2.


OK.


You know what I mean


OK


So when I say first two is that these two, right? I'm just trying to write here. So you'll see the blue color pen.


Uh, yeah, so the reason is they result in


uh in similar


factors which can be canceled out.


simplifying the expression.


which can be pulled out as a common factor, not cancer yet, but pulled out of a common factor.


or yeah


It's like the


because what will happen if you do the 5 X, then you will get like a random 5 x 4 or something.


And then if you go on the other side, you will get like a random, uh, what is it? Again, again, a fiber. Actually, that works actually, I think, right? No, no, you get a 5 by 4.


You don't get a 5x 4 there. You get, uh, what?


2, you get something else altogether. So you get


you kind of stalled


Actually, let's try it. what happens? I never tried it.


Well, well,


You ask me some questions. OK, so you put this 5 x 4 on this side.


And then I get a 10 x 8 on the other side.


OK.


Wait, a question with the negative 2 though, before we do that, um, so the negative 2 you can basically do what exactly we can basically like cross it out or?


No, no, no, you cannot, you should not cross anything out. You're factorizing it, so you take that factor out.


So, like you have the negative2 in both, meaning what exactly like.


Are you saying, uh talking about the X minus 2?


Yeah, cause that's in like both of them. Like do


Yeah, yeah, so you take that X minus 2 hours. I'm writing it where your cursor is.


and then you put the rest in the bracket.


OK, so you can do X minus 2 and then 4 x plus 5,?


Correct.


OK


Yeah


And then you'll get it to that form again, yeah. OK, so it turns into


And then all of that equates to zero, which you have not written anywhere yet, but the idea is to


find the factors


births to X minus


X minus 2, and


4 x plus 5.


Forest is 5.


but


my word be like


to


X minus 2


+ 4 x + 5 cause there's, or right now.


It's like because you're


you have two of the times x minus 2 in there. So I was just


confused like


how you deal with that


Yeah, you take it out, this and this.


and you put it under this umbrella right here.


Yeah, but like


And then one greens is the + 5.


that is here


And then what remains is the horax.


So now if you multiply it, it'll be like, you know.


X for x square, right?


OK. Yeah, I was just asking like why.


it would, like it just, like it would result like when you combine the X-y this twos together, it would still be x minus 2.


And, and


No, I was just confused a lot.


I don't understand. So why are you saying this X minus 2 will remain as X minus 2 here?


Yeah, cause you're, cause you're like combining both of them.


So


or


forget about the X minus 2. If this was 8th year.


Mhm


if X minus 2 is 8 and X minus 2 here is 8.


Would it change


Oh, so then we're just saying basically like that's


you're doing


Think of it just like a number. Yeah, I just.


like it's like why


right?


Yeah, exactly. You just pull it out. That's it.


So you're basically doing 4xs Y + 5Y.


Correct, exactly


OK. Even then though, like you combine them, I was just like confused like where the other why goes cause


OK


By y plus something multiplied by y. I was just like confused cause and the second one you're doing only one thing multiplied by y, not


Oh wait, wait, wait, because you're multiplying both of them by y.


Exactly, yeah, yeah.


Yeah


OK. So, OK, here. Yeah, you're just basically combining them because you're just doing


like 4X for example, let's just say that's 4 and then 5.


is


Yeah.


Yeah, it's like basically doing this to uh to multiply it by 5 + 3, for example.


Yeah, like just


So 52 is 10.


And plus 6


and 6. Yeah.


And then in the other one, if you did the same thing


wait, uh, whatever.


Yeah, yeah, this is 10 + 6 is 16, or you can do 5 + 3 is 8, multiply by 2 is 16. Both are 16, both are 16.


makes more sense now


And then


defined X, you would just


com, you would assume.


uh equal to zero. So


right?


Yeah


like you would assume X minus 2 equals 0, right?


Yup


Each parenthesis


equals 0


X


uh


X minus 2 equals 0.


So then that would mean X


equals 2


And then for


uh, 4x plus 5 equals 0.


then X would be.


5 equals maybe 4 X and now be just 1.25.


Oh wait, no, 1.125.


Wait, but you have negative 1.25 again, no?


from before


or is it unrelated


Cause you know how like from before,


um I got 1.25.


Yeah


like 1 point, yeah.


Is that like


Yeah, yeah, so that should work because I was gonna say that, yeah. So it shouldn't matter, but uh


which just makes it more complicated.


it'll make it more complicated, correct.


OK


Um, I'll put that in the orange.


because what we'll get is you'll get like a random X + 5 by 4 X something like that.


But yeah, that should work though. It should not give you any any other uh answer. The answer should always be the same, then how you approach it.


OK.


Actually


All right. Hopefully that was helpful.


yeah.


We just need a couple of more practice problems and you should be good to go.


That's like the homework that I was, that I have later.


cause I just didn't get it and I was like, oh my God, I, I don't get it.


Oh, OK. So this is, is the phone book or is it still the class work? You still, you do the homework?


Uh, like a class work.


Well, this was the homework


Yeah


No, no, no, this was like the classwork a few days ago where they modeled that, they modeled out how to do it, but I didn't get what they did, so.


Got it, got it. OK, got it


Um


But go through the link if you forget it, the link I shared, I think that should also give, you know, additional color and some practice problems you like.


OK. But also, like how would we use the table thing like with the cross on.


or is that or or the way we did it, it's like you can't really use it.


The table thing is also trying to find factors.


that


No, no, no, like the cross, like the, the one at the top where it's like 3 X plus W times whatever.


Mhm.


Like, um, how would we use legends that


Oh, you mean the graph


Yeah, the grasser


Or is that just like


I don't know


what we're doing there. Oh, you're, you're talking about the area method.


The what now


So there is something called the area method, so let me


show you what. So your answer is X minus 2 and 4 x + 5, right?


Uh


This, this, uh.


Oh yeah, yeah, yeah.


Yeah.


check, check by my markers I'm gonna write something in blue. This one, right?


uh Wait, where are you at the bottom?


Yeah, you scroll down to the bottom.


OK.


See my cursor?


It's just got more lay over time.


It shouldn't


Oh, a computer slow.


Uh, yeah, I have at the bottom.


Yeah, you see my cursor, yeah, good. I see yours now. OK, now, so what, so the, what they're saying is essentially this, uh, bear with me for a second.


Dear.


So think about this as 4 rectangles. Do you see it?


What they're saying is that one rectangle, you put X, X, and then you put 2 here.


Like I don't have like a basic area model works, but


Oh, that's what they're saying, I think. No?


but none like the way they're doing it is like they're converting it into this form and that format. I just don't really get why they're doing with some of that.


I don't know, what you think.


or even like what it does.


Yeah, I don't understand what they're trying to do there then maybe if you're talking about this on the rough work, they did some, you know, factors. Do you see that?


The 36 is 36 multiplied by 1, 18 multiplied by 2, 12. Do you, are you asking that question?


No right


No


I, you.


I mean, I'll just, I'll just ask my teacher if it's fine if I don't use it because the way you told me isn't that complicated. It's


uh


Yeah, I, I don't understand what he's trying to be trying to communicate there.


OK.


Well, I think that's it then.


but


So I have to do the homework


a look


All right, good luck with the homework. I think this should be easy now that you have this.


Yeah


All right, have a good evening


Yeah, bye


Hello


Hello Emily, how are you?


I am fine, how are you?


I'm fine thanks


so we started a topic called higher order derivatives and graphs and I'm kind of confused, so I wanted to see if you could maybe help


do you have a question that you are working on?


Yes, but do you know what they are?


Yes, I know what higher order derivatives are.


okay, I'll put something on the board


first question


how do you find the derivative of e with that big number 2470


what is f' ?


not f' f''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''


basically f(2470)


I am asking you to first find f'


do you know how to do that?


OK, what is f"


Yes, but let's write it a bit differently


What is the third derivative?


what is f(4)?


do you see the pattern?


is all you are trying to say is that whenever you are trying to find such a big derivative for the power of e^smth, all you do is keep the 2 and just put the big number to the power of it?


I am asking if you see a pattern?


yes, that's what I was explaining


good.


Any questions about that?


no i guess we can move on


there is the next one


I wrote the answer, just to make sure.


ok


Is that the graph of y=f(x)?


i guess


this is a different question


It should say in the problem. Is it the graph of f(x) or f'(x)?


it's f(x) since in the answer it's asking for f'(x) and f''(x)


Well, that's not actually true, but let's assume that it's f(x)


Assuming that the graph is f(x), is f'(a) >0  or <0  ?


i wanna say >0


You don't sound very sure. Why is f' >0 at that point?


i am sure, it's bc the slope of the tangent line is increasing and looks higher than 0


You mean that the tangent line has a positive slope.  That's right.  Also, even without thinking about the tangent line, you can see that the function is increasing at that point.  If a function is increasing then f' >0


How about f" ?


but on the notes he was confusing us saying that for example(watch what I'm doing on the board) is increasing


how to know that a function is increasing?


He is not talking about the function increasing.


we will get to that.


at the blue point, is the curve concave up or concave down?


Right now I'm not talking about the concavity yet, I'm just talking about that portion, but ok


down


When a curve is concave down, is f"  >0 or <0 ?


that Idk sorry


OK, you haven't learned that yet.


as I said, he did talk about it, but I'm js so confused with his explanations


I drew three tangent lines, A B and C


ok


what might be an estimate for the slope of A, just roughly?


-2, maybe?


OK


How about at B?


OK, so is the f',  the first derivative, increasing of decreasing over that interval (we would call the curve there concave up, because it is facing up


Again, as we go from A to B to C, is f' increasing or decreasing


it's increasing


right.


Now, f", the second derivative, is the derivative of f', right?


yes


so if f' is increasiing as you go from A to C, would you expect f" to be >0 or <0?


to find that out should I draw a second graph based on the estimated slopes we found or is there an easier way?


the point is that the derivative tells us if a function is increasing or decreasing.  f' is itself a function.  If you tell me that f' is increasing over an interval, then that means that the derivative of f' should be>0, and the derivative of f' is f".


so if f' is increasing over an interval, then f" on that interval would be >0


so that make sense, if f' isincreasing on that interval then f'' is also increasing on that interval?


Exactly.


no, sorry


f" is positive over that interval.


wym no sorry?


so f'' is not increasing?


f">0,  being >0 is not the same thing as increasing.


Using a similar argument, we can see that f" <0 where the blue point is, where the curve is facing down.


okay how to know smth is positive vs increasing


In this case, they just want to know if f" is <0 or >0  at the blue point.


I'm asking just in case in the future I'm asked that


Something happened to the site. Are you still there?


but since you can see that on the left f" >0 and on the right f"<0, then that that means that as you move from from A to the blue point, f" must be decreasing


Hello?


sorry


I got disconnected for a long time and js got back :(


Do you understand this question now?


okay so let me recap and tell me if I'm correct


so where the blue point is f' is decreasing because the slopes around that curve are going from estimate 1.7, 0 to -2, and this then means that f'' is <0


yes, that's right.   It's the fact that f' is decreasing (which means f" <0) that gives the curve that concave/facing downward look.


concave down/facing downwards


does being concave up or down say anything at all?


They describe the shape of the graph.  That's something you couldn't do as well with just algebra


what I'm asking is does the graph being concave up or down tell us that it's increasing, decreasing, positive or negative?


at all?


It tells the shape, as I drew on the board. Do you see that?


yes i see


It wouldn't tell if the function is positive or negative.  the curve could be above or below the x axis. it could be anywhere


ok


another question i have is f' or f'' being <0 or >0, does that say if it's positive or negative?


Numbers that are <0 are negative and numbers that are >0 are positive


Are you all set now?


okay that makes so much sense thx


Do you have to go or you got a little bit of time?


I can do another question.


okay thx one min


what do you think?


hello, are you there?


sorry


Do we have to create our own graph?


You can, but it's not necessary


okay, so what do we do then?


First of all, is f'(2)) most likely postive or negative?


maybe positive??


what does a derivative >0 mean for a graph?


increasing


Yes. It is important to remember that.  Does it look like the function is increasing?


no it's decreasing


Yes.  For a fn to be increasing, that means that as x increases (as you move left), f(x) would go up,  But here, it looks like f(x) is going down, so f(x) is most likely decreasing.


yes


We can estimate the slopes of the tangent lines at x =2, 2.1, 2.2


we can do that by figuring out the rate of change between the points that they give us.


What is the rate of change (the slope of the line) between (2,3) and (2.1,2.7) ?


That average rate of change gives us a rough estimate of the slope of the tangent line at x =2


Now let's move to the right.


let's get the rate of change between the next two points


One more


Good work.  So, does it look like f' is increasing or decreasing over the interval from x =2 to x =2.3?


it's increasing


Yes


so will this curve be concave up (f">0)  or concave down (f"<0)?


this will be concave up


that's right.


I also plotted them on Desmos, so you can see.


I think you can see that the curve looks concave up.


yope


to recap, first we found out that f(x) was decreasing, then we looked for the slopes and found out that f'(x) is increase which led us to the knowledge that f" is concave up or >0, correct?


yes, that's right


We used the average rate of change over small intervals to estimate the slope of the tangents lines


okay now questions time, first question why did we need to find out if f(x) was increasing or decreasing?


because the question asked if f'(2)) most likely >0 or <0.  If f(x) is increasing that means f'(2)) >0 and if f(x) is decreasing that means that f'(2) <0


wait so the shape of f(x) helps us know the answer for f'(x) and the shape of f'(x) helps us know the answer for f''(x)?


The shape of f(x) helps you to know both f' and f".


if f(x) is increasing over an interval, then f' will be >0 over the that interval.


if f(x) is concave up over an interval, then f" will be >0 over that interval


You don't have the graph of f' ( although you will be given graphs to f' to analyze at some point).


so if f(x) tell us both f'(x) and f''(x) what will the graph of f'(x) help with?


You will eventually get questions asking you to analyze graphs of f'.  I wouldn't worry about that right now.  A graph of velocity is a graph of f'


oh okay so, just tell me everything i need to know then, and then I'll let you go, When I have either a graph or a table, what are the steps I should take in order to find the right answer?


I am not sure what you are asking.  We just did a question with table.  If you look at a part of a graph where the curve is concave up, f" will be >0 there


i was basically just asking you to recap our session in your own words meaning, just tell me first thing you look at is blah blah then it's this and that and to finish this and that, that's it


The most important thing is to remember that the derivative is a slope, the slope of a tangent line at a point. So if a fn is increasing, then f' will be positive.  if f' is itself increasing over an interval, meaning the slopes are increasing, which means the second derivative will be >0, and that makes the curve look concave up.


This should all become clearer as you do more problems.


okay thank you!


You are very welcome.


Have a good rest of the day!


YOu too!


bye


hello


Greetings, Mr. David!


How are you doing today?


ok,!,..what are we doin?


Awesome!


So, I have a Geometry assignment I have to complete.


It shouldn't take that long.


It's only a few questions.


famous last words


being funny


I did the rest of my HW earlier.


oh


haha


May I post my assignment on the board


yes


Mr. David


also


just letting u know


my computer may freeze at times


my head may freeze too


(It hasnt happened so far)


oh lol


same


so i hope it wont happen


and when my computer freezes it takes me like 10 minsfor it to work again


ok


one sec


so before we assume


my teacher wants us to do odds (thank god)


k


whats the question on 43?


so i have to do 43, 47, 49


and 53 and 54 (will post after)


Mr. David


so 5 problems


thank*


oops


yeah


the question


Determine which line passing through the given points has a steeper slope.


thats the question


ok so lets find the slopes of each using the y2-y1/x2-x1 method


can ya?


yea


this slope is horizontal mu m=0


no


nevermind


its not


haha


its 1/4


for line 1


for line 2


3/7 is the slope


line 2 has a steeper slope right


yes, 3/7 is bigger that 1/4


yay


assuming you did it right, ha


i did it mentally lol


im in Grade 10 honors


ok


im confused on 47


hows about 47


ill set it uyp


ill find the slope on the first line


ok and set that equal to the next line slope form


i dont quite get it


ok, see board


yes


i did that off the top of my head


yes!!!!!!!!!!!!!!!!!!!!


YAY


same thing for question 49


could you show me how to do the work


only now they are negative reciprocals


k


instead of doing it in your head


i simplifiy first


first ill get the slope of the other and flip it and - it


oh wait ,...we are doing the two //


i was looking at 48, duh


ok


ok, so we gotta make them equal


if you solve for y,www.mathsolver.com get y = 0


see?


yes


just did the algebra to get y=0


i thought y was 8


tricky lil question


im confused


how I got zero?


yea


and how you got 3/2


made y=0


ohhh


and 0 +6/4 = 6/4


i didnt see it till i solved for y though


need me to show you that?


but i also got 8


is the answer 3/2 or 8


ill do it on board


cross multiply


`ok


oh


so 2(y+6)


ohhhhhhhhhhhhhhhhhh


i get it now


and then u plug 0 back in right


yup


and also


how did you get y--3/3-1


and y--6/9-5


the slope form of the lines


==slope


ohhh i see


we ok?


so its 4,3/2 right


they want the value of y


then what do we do with 3/2


and in the other equation -8


just the two slopes have to be equal to be parallel


if y is zero, we get equal slopes


ok


so y is 0


ok and thats the final answer


yes, you sound like the guy on "Who wants to be a millionaire"


HA


what else,...we got


and 54


ill post 53 and 54 after 49


we just did 49


k


we did?


i thought we did 47 just now


im so confused


we did 47 above


you found it y to be -8


oh


and 49 is 0 right


hey Im supposed to  be the old one losing memory


yup


im just tired


lol


i gotta go soon


bro i overslepped and skipped 25 mins of class today (biology)


anyways


we are almost done


I think Mr. David is right


becaue he did y2-y1/x2-x1


and Ms. Green did y2-y1/x2-<US_DRIVER_LICENSE>


which is wrong


riht


right*


yea


yay


and last one


i can do 54a with a grphing calculator


one sec


coordinates of B and D


I have it on board


(2,4) is b


(10,-4) is d


right


horizontal line has slope of zero


vertical lines have undefined slope cuz we cant divide by zero


are we good?


cool


good luck!!


what do u mean good luck


just a phrase


oh i thought u were leaving for a sec


i thought we were


is a correct


the points i got


yes see mine also on board?


i did not notice that lol


ha, once again,...good luck!


so how do we show that ad is parallel to ab


nevermind


i meant ad and bc


and ab and CD


remember, we show the lines are //


same slopes


so both of their slopes are 0


or yeah


have the same slope


as I said,.....if you do the slope form


and horizontal lines slope is zero


just do the slope form for the // lines


same thing we have been doing


ok


and c


final question


show the perpindicular lines have slopes that are negative reciprocals


AB perpindicular to BC


vertical line undefined slope


but if you flip the verticl line slope over its not dividing by zero anymore, its just zero


and zero is slope of horizontal line


thats all


oh ok


and thats it


yup


yayyyyyyyyyyyyy


thank you so much for helping me


:D


Have a great night! :)))))))))))))))))))))))))


sure,adios


ditto


Ciao


ill leave u a good review


cyaa


bye


Triangle S has an area of 50 square units.
Ben drew a scaled version of Triangle .
He labeled it Triangle T (shown here).
What scale factor did Ben use to go
from Triangle S to Triangle T?


Hello


hello


can u please show me the problem


the problem is  Triangle S has an area of 50 square units.
Ben drew a scaled version of Triangle .
He labeled it Triangle T (shown here).
What scale factor did Ben use to go
from Triangle S to Triangle T?
this


isn't there a drawing?


imma post it


k


I haven't see anything yet


im trying to uploAD IT my computer isn't screen shotting


what kind of computer do u have..a chromebook?


lenovo


perhaps u can draw it


ok


can you help me know


now


do u have any dimensions for the smaller triangle?


this is the question btw


i dont have


so can you help me find the anwser or


so is there a picture of two triangles or one?


there is only one on the assignment


and 50 is the only number you are given?


so triange T has a height of 5 suare units and a width of 4 square units


yes 50 is the only number


ok..u know the height and width are important for this problem too


ik


do you know the formula for a triangle?


no


im in 8th grade


do you know the formula for a square?


how do you find the area of a square?


multiplication


of what?


say if you have two sides, a and b..what is the formula for a square?


that how u find the area through multiplication


what a formula


can you explain


what is the area of this square?


Whatat I drew


the square is the question


i dont


a x b = the anwser


correct


oh


so i have to find the two number that give 50 on the question


would that be 10 x 5


if u cut up the square into two..what do you get? ....what shape?


rectangle


Thatat is one shape


say if u cut the square diagonally..what shape do u get?


triangle


so how am i solving this question Triangle S has an area of 50 square units.
Ben drew a scaled version of Triangle .
He labeled it Triangle T (shown here).
What scale factor did Ben use to go
from Triangle S to Triangle T?


please be patient


ok


so two triangles make a square right?


yes


so if u cut the area of a square into   two...u get the following formula (a*b)/2...which is the formula for the area of a triangle


h x w


so what now is the area of the triangle for the dimensions u gave me?


50 for the first one


20 for the second im sure ?


is that correct


no


using the formula I gave u ..what is the correct answer?


it would be 10


exactly


very good


oh


so the sf would 10


now the bigger triangle has an area of 50 and the smaller triangle has an area of 10


how many times is bigger one bigger?


no the smaller traingle should have a area of 20


for area you do height times width which was 5 x 4


there is no divison


why should the smaller triangle have an area of 20


remember, the formula for a triangle is (a*b)/2


5 x 4 is the area of a square


we don't want the area of a square


no it isnt


Triangle S has an area of 50 square units.
Ben drew a scaled version of Triangle .
He labeled it Triangle T (shown here).
What scale factor did Ben use to go
from Triangle S to Triangle T?


the picture i have


states  5 x 4


hello


yeah...


height is 5


so what sf did Ben you


width is 4?


yes


Ben use ?


what is the area of the smaller triangle?


the smaller triangle area is 20


and the bigger triangle area is 50


height x width will give u the area for a nott a triangle


so u have to divide by 2


oh mb u right


so what the sf now


what is the area of the square that I drew?


but it is a triangle with the same numbers


since we are dividing it it would be 10


so the area of both shapes is 10,50


it doesn't mean they can have the same numbers


ok


if I cut the square in half and ask you what the area of the shaded area is..what would it be???


correct


ok what is the scale factor Ben uses


so we are looking at half a square which is a triangle


yes


so how many times do I have to increase the smaller area to get to the bigger area


for the 50 area triangle


how many 10 (area triangles) fit in a 50 (area triangle)?


so your scaling factor would be 5


the bigger one is 5 times bigger than the smaller one


make sense?


thx


your welcome..enjoy the rest of your day.ep up the good work


make sure you understand the difference between the area of a triangle and a square


hello


Hello, how can I help you today?


i have a homework assignment i need help on


sure put it up whenever ready


which ones do you need help with?


all but first 1


everything except number 1?


i meant lets start with one


ok then


do you know how to solve for slope?


y-two - y-one / x-two - x-one


yes


ok then for number one lets label our points


2 is our x-one, 6 is our y-one. 5 is our x-two, and 1 is our y-two


so you would do 1 - 6 / 5 - 2


and our slope for number one would be -5/3


it says simplify


yea so for number one thats all you can do. you cant simplify it anymore


oh okay


should we move onto number 2?


ok so for number 2


3 is our x-one and 8 is our y-one. -1 is our x-two and -4 is our y-two


so you would do


-4 - 8 / -1 - 3


to get -12/-4


and when you simplify it you get 3


so the answer for number 2 is 3


how do u get 3 from -12/-4


-12 divided by -4 equals positive 3


negative and negative equals positive


okay


should we move onto number 3?


yes


ok then for number 3


5.4 is x-one and 2.9 is y-one. 3.7 is x-two and 4.1 is y-two


4.1 - 2.9 / 3.7 - 5.4


to get 1.2/-1.7


any simplifying?


probably mot


number 4


ok then


for this one


2b is x-one and 2c is y-one


b is x-two and c is y-two


so we do c - 2c / b - 2b


to get -1c / -1b


then simplified it would be


c/b


because the -1 cancel out


okay i understand


number 5


5 is x-one and 5 - t is y-one


3 - t is x-two and 4 - t is y-two


4 - t - 5 - t / 3 - t - 5


which becomes -1 - 2t / -2 - t


which simplifies


to


1 + 2t / 2 + t


because the negatives cancel out


that one is confusing


yea it has a bunch of variables but it is about the same


yea


number 6 then?


yes


5 is x-one and 7 is y-one


17 is x-two and 7 is y-two


7 - 7 / 17 - 5


which is


0 / 12


and if you divide anything by 0 it is 0


so for this one simplified


is 0


okay


ok


ok then


6 is x-one and 0 is y-one


15 is x-two is and y is y-two


so you do y - 0 / 15 - 6


= 2/3


how is y-0= 2?


we are solving for y


so you have the equation of y - 0 = 2


y = 2


once done solving


so the answer for 7 is y = 2


nvm


y = 2 is the beginning


since we know that 15 - 6 = 9


and 9 divided by 3 is 3


we need to multiply the 2 by 3


to get 6/9


so y = 6


im confused im sorry can u write the equation on the assignment i sent u


Hi


Hello. Hi, Michael


I


Well, not on the face. Hello.


OK


Good. Uh, do you have a, a certain question for us to do here today?


Uh, yeah, I do have a question.


Yeah, go ahead, Michael. I'm listening to your question.


So it's like a problem about um applying properties and like uh uh additive inverses.


Excuse me, I was just unmuting the mic. So can you just let me hear that again, please? Run me run it by me again. Thanks.


So, uh, I have a question about like applying property stuff.


Mhm


OK, good


and uh


Can you write down your question, please.


yeah. OK.


Yeah, and what happened to the question? Mhm.


Yeah, so I have to figure out the reason why you have to uh use uh why I have these additive inverse before we like properties.


Yeah, yeah, that's good. And have you got a question that you've been given to answer with you now?


Yes.


Oh, lovely. Good. Can you, um, maybe put it on the board for us? Is it, is it possible to put the the image up for us to see or copy and paste it into the chats, it's fine.


Yes


That will be grand


There's the question


There's the question, isn't it? Let me have a look now for it. Is it on the board?


Oh yeah, yeah, it's just about loading for me now. Thank you.


I'm gonna scroll up


Lovely. So, um,


Michael, I'm gonna ask you, um, did I say Alex? Emily, sorry, but could you read out the question for us, please. Thank you.


Oh, OK then. So one use the expression 6-9-12 + 22 defining his profit for days 2 and 3. He rewrote the expression as 16 + 9 + 12 + 22. he can use the associative in the community of properties where we write the expression again. Explain why he had to use out of an inverse before she could use these properties.


OK


Um, so you're not really asked to do an answer, but just to explain the answer then.


Yes, so we have a hard time.


Yeah, I'll check, yeah.


Do you have a heart


I have a hard time explaining the answer.


I'll explain you so good, good, good.


All right. Do you have any, uh, notes on this topic that you'll teacher's been going through with you?


Yes.


OK. Would you mind, would you like to just read us out a portion of that, just the part that's connected to the question not only.


Uh, OK. So the additive property is subtraction and community of property, A + B equals B + A 5 minus 44 minus 51 it's not equal one, but 5 plus 4 plus 4 + 5 equals 1 equals 1.


Oh


And so, yes.


That's really a lot of notes you've been taking down. Well done, um, Michael.


Yeah


OK. So


OK, so


what was the


additive inverse that was used in this question. Could you give us one of the examples of an additive inverse in this question.


Um, so like I didn't she's basically just the opposite of just the opposite of a, of a quest of a question like example.


of a value


6 minus 9 equals 6 + -9.


Yes, yes, good. So that's a good example. Yeah, could you use the question itself then to give us another example of an additive inverse now


OK, yeah, so 9 minus 12 equals we, what they're getting 9 plus.


So


uh.


Instead of saying


add


a 9


Is that right? You need to use 16, add the negative inverse of 9.


It's just an example


Oh, we, we had the connection timed out for a bit. So I can just about hear you. You were saying, what's the example again?


The example 16 minus 9 equals 16 + -9.


Yes. So, you have used


an inverse of 9, which is -9.


Yeah.


Yes


And why did it, yeah, that's it. So you've got the answer. Why didn't he use just uh straightforward 9, and he


um, to do the quest to to get the expression correct? Why did he have to use, um, the neck, the, the negative of 9 or in at the inverse of 9 instead of straightforward and a 9 on its own.


Um brex and best sense and yeah.


because that's what they want to know


Oh, OK


is because it was like, it would sound confusing for the question and plus, so they would have to simplify


in order for it to fit the actual question.


Mm.


Yes, in a sense, um, what's the meaning of the word inverse?


Michael


like the opposite of something.


Yes. So, what's the inverse of


uh, profits that you make.


What's the


reverse or the inverse of a prophet.


He reversed, she's like the debt.


Prophet


OK


A debt, that's good. Debt. So they're saying that if you had


written + 9 that would be referring to


a prophet


not a debt


Oh no


because you need to show, yes.


Yeah, I, I know now.


Because you need to show a


You got it, yeah. You, you, you need to show that


a that, that, that's it, that's it.


which is the inverse of a profit. Is that right?


Yes.


Yeah, that's the reason for the negativity. That's the reason negativity so coming up or the additive inverses we need.


Don't add on


I'm not back yet. Oh, because of you.


Yes


If you didn't, if you use um the number without


the um inverse sign. We're talking about a profit, and we don't want to be talking about a profit for those values. We want to talk about that, debt or a loss.


Yay. So, you got that one right?


Brick.


Right.


Oh, that's lovely. Uh, can you tell us, uh, have you got any other questions apart from this one? You, you can, OK, can you go back over how, how do you explain the answer? Why didn't he have to use an additive inverse, instead of a straightforward value, um, positive value.


Now that's gonna get you and it's just the same way.


You're looking.


Oh sorry, um, sorry, David, it's because, um, uh, it's because like it's supposed to represent dead and not like an increase in that you.


I like the same value.


That's correct.


Yeah, yeah, because if you'd use the same sign, it would have meant you were talking about a prophet, not a debt or a loss. That's the reason. Because you need to use an inverse


on opposite sign to show that you're talking about that. True.


OK


OK, let's


Well done, well done, Michael. Great work from you here today.


Yeah


And um


could you tell us a bit more about what you've been doing on this topic.


Right


Mhm


I have just been eating a I, I have been um doing like, like how I like to track like, um, uh, integers and stuff.


Yes, right. So can you give us a little question on that um section that you've been doing as well.


For example, uh


Yeah, give us some other questions. OK.


a 5 minus 3, and I would have to use a number line, and if I use the number line, then it would be it would be.


negative -8.


That's correct. Yeah. Could you draw a number line on the board, please?


and show us -5


minus 3 on it


Yes.


Thank you. That's lovely. You break.


So, uh, 5 is over here, -5. It's supposed to be over here then


if I move it by 3, it would then be able to truck. I'm like just adding -5 to 3, which is -2 wait a minute, wrong way. I'm sorry about that. 123. It will go -8. This means -5-3 equals -8.


It's basically just addition but going the opposite way.


I I think


Hello, are we back on?


Yes


you know


Oh, that's lovely. So you were saying that to do, you're doing another question just then.


Which question was it? You were doing work


It was 5 minus 3.


-5 minus the same.


OK, yeah, I got you.


No, -5 minus 3.


OK, good. I think you put something on the board for me, let me see what it is. OK.


OK.


Uh, yes, I mean I see if I can move the canvas of this, yeah, I can move it to. So I can see, yeah, you've got your jumps going on, your, what is the um direction


um


Oh.


on the, on the number line. Um how is it linked to the sign? How is the direction you're traveling linked to the sign of your number.


Mhm


so it really depends on the sign, like if it's an addition, then you move right because the right represents the positive side. And if you move towards the left, you, it's like subtraction because you're moving towards the negative side.


That is so good. Well done. Thanks, Michael. That is so good. So you've understood about um how the numbers are linked to the direction of travel, the sign of the numbers in, yeah. So a negative3 means you're going 3 units to the left, like you said.


And uh yeah, that's great.


So you've shown, you've shown me 3 jumps to the left. That's good for -3. And, of course, -5 means you need to go another 5 jumps to the left, and you'll end up doing 8 jumps to the left in total. That is so good


which gives you the right answer of?


-8. Let's see if we can, um, change some colors on here. Yeah. I'm gonna put a big tick for you.


next to your, if you'll let me know.


do that good


Yeah, thank you for that. Lovely work.


And what um


sort of exams are you're going to be doing?


this time, do you know


Um, maybe it's, maybe an exam that includes uh subtracting integers and maybe finding out like, like I was crying like the distributive property and like not not sur for communities or associated properties.


I say a prophecy as there we go.


So I want you to use some algebra on it this time.


we have um


say, minus, can you write on the board, minus 8A.


plus 15A.


and give us an answer


Uh OK


So this is your question. We, mhm.


A a.


Wait, 15A. So, uh,


-8 a add 15A.


is not.


OK, I, so I got the, I got the equation on the board.


Mhm. Let me check for it now.


on top of that you get So the answer


so uh


Yeah, yeah


vocal the vari variables are A, so it's, so it doesn't really matter. The very, the variables don't really matter at this scenario. We just have to focus on the, the integers.


Yes, yes


So then, uh, parking.


Uh


but you have a So if I use the number line.


I'm gonna have to clear all this out.


I'm gonna have to cut all this off.


Yes. You can


So for you.


Yeah, that's fine. I've seen that and it's all good.


by uh


Mhm


Mhm.


use the number line then uh I will move neg if I will move apes units to their left, and I would move 15 to the right, and if I do that math, that's the same thing as 55 minus 8.


So then, um, 15 minus 8, that is equal to


it will, it will be equal to 7, so the answer would be 7A.


Exactly right. Thank you. This is good. Thank you for that. Um,


Michael. So can you try us a bit of the distributive method as well. So we're gonna try


the brackets method


Yeah, can you do me the answer to -3.


open brackets


plus 7


OK


Well, wait, hold on, there's something let me write on the board. Oh, wait, never mind. So 8A.


Yes, start with a -3 in front of brackets.


to stop and do it.


Oh wait, um, sorry about that. Wait.


OK, so 3 is now on the brackets.


Yeah, it needs to be actually in front of it. Let me see if I can write it as well for you.


Yeah.


The minus 3 in front of the, and then you open the brackets after that.


Yeah, and then you can finish writing the question


for us


So what was the question again?


Yeah, I've put the question on the board for you, but.


In green writing.


Can you see it now


I can see you on the board now.


Uh, yeah.


Hm


So, you understand the distributive property as well, because I, I can see you've mastered the associative and commutative properties quite well.


Good.


Good


Yes, you know how to use the just distributive property as well. So it's like -33A plus 3 times 7.


Yeah. Can you simplify for us?


Well, um,


OK, I can try. Well,


Mhm


Just remember to do what you had to do with the first question you, you had, you put up on the book. You might need to blend some of the signs together when you simplifying a plus times a minus can be blended together.


OK


And the brackets is shorthand for which, um, operation.


You can actually replace brackets by


which operation or sign?


Um


uh


A time sign


Hm


So you could actually put in


minus 3 times 3A.


yeah


Yeah, that's what you could replace the brackets by a time sum. Good. And the same happens with the other term plus minus 3


times an 7.


I


Now you can simplify for us. Thanks.


What is the answer for minus 3 times 3A.


OK, what we'll do is we'll, we'll, we'll do another easy question to start with.


Right.


oh OK


Yeah, we'll do some plain numbers.


Yeah, we'll do some pain numbers. So this time, can you do minus 3


times


brackets


+ 32.


Can you give us the answer?


to -3


in brackets 12 + 32.


Yeah


Very good


OK, so first what you gotta do is you have to find out the meaning of of three square, which is uh 9, so it's 12 + 9, and that is equal to 21 and 21D 3 times 3 times 21, uh equal to uh um


um.


hold up


Yeah, that's going well. Thank you. Mhm.


It's, it's probably equal to uh, I know.


basically just 21 times 3, but negative.


Yeah, so you can add up 21 3 times. Can you add it on them onto each other. 21 + 21 is?


Uh, it's 42.


Good. And 42 out of the 21 is


Thank you, that's really well done, Michael. Have you got any questions in your, uh, school book?


on the distributive properties for us to do.


Yeah, give us one moment


Well, I do have questions, but uh they're already answered. Just that I do have


Oh, there you know good, good, good.


Mhm


yeah, they're already answered.


OK then. Thanks so much for being such a, a great, great, um,


proactive student today. And do enjoy the rest of your, uh, studies in free algebra.


OK, bye.


I don't know.


Bye


one


I didn't like that


Hi


Morning! How are you?


I am well


how are you?


How was your weekend?


I'm good. It was good.


how about you>


It was good. It was good, just um


was calm, just what I, I prefer.


Excitement gets to be too much.


Awesome!


Man, I am so out of practice.


these intervals


integrals.


Oh, the, what did I do that?


Anyway,


um


have you been able to


or let's see, where, where did we leave? Actually, did you want to start with anything?


Any comments or questions or


hope I didn't lose you


let's start with last topic


where we left you


Where we left me


ohh, sorry, off**


let's see, where did we leave me last time?


Right


So, last time we had glossed over.


the various


um forms of integration by fractions.


mainly this table.


which just covers like the form of the restroom, but you notice that they don't really go beyond um 3rd degree polynomials here.


Um, that is because


you can typically


uh


rewrite


the expression


um for a higher norm for a higher-order polynomial.


If you can, if you can't if you can factor it and then kind of rewrite it as um


a group of smaller polynomials.


or lesser order polynomials that preach you the best way to go.


But I think, and I also don't think they're gonna give you


they're gonna have you do problems that are


higher order than 3,


right,


right, but I, I don't think we actually did any problems.


Um


let's see, what do we cover last time? We had finished doing


the particular


um the previous chapter covered


particular integrals


which involved


um


just in just trig properties.


and logarithmic properties, right? We had this table, which I really wished they would have gave this its own table.


rather than the way they have it written out.


yeah, that would have been very helpful


Uh, but again, these are all just various um


various forms.


that I borrowed that are already known, they've been proven, we've seen them proven. Um, they just kind of help the integration process.


right? So that we, so that we can get whatever problems we have, we can just, if we can figure out how to write them, um, typically by substitution, if we can figure out how to write them in the forms that we know, basically, uh, um, these


uh


forms which are, uh, derivatives of either 2nd order or 1st order.


um, polynomials


Well, square rooted second-ordered polynomials.


Um.


then these already have known solutions.


yes right


and then it just becomes a form of, you know, how do you apply it?


right? And then now we get to, well, what happens if we have


um rational


expressions with


polynomials and denominator, as always, partial fresh, where um


they do, they have a B term in the middle.


right, because the previous polynomials were always x-a minus x, right, which could be


factored into um X + A or X minus A or so on and so forth.


but never really, um,


the full expanded form didn't cover having an uh


an X term in the second-order polynomial. Now we're covering that other aspect where now, OK, we have


a second-order polynomials that can be factored into two different expressions, but have this um


what is this? A + B term?


for X


or even 3 or polynomials.


Um.


which I think actually


the last few polyno the last few expressions also covered that.


Where is it


Yes, here


Yeah


wild


Wild.


Anyway, so logically


you can kind of see um the progression. It's just typically as these things go, that, you know, integration is a skill.


and only we get better, if you get better at a scale is to just keep practicing.


So


that is


kind of what I've planned today.


um, unless you've already practiced, done problems. We can go over the example problems


Um,


if not, then we can move on to


if you have done them, then we can move on to um


I did 7.4


integration by parts.


You did 7.4. OK. So you haven't done 7.5.


So we can go through the example problems and then move on to actual problems. And then, um, touch on, actually, you'll probably have time to do integration by parts depending on how


many practice problems we do.


Yeah


Yeah, and I need to practice. I was trying to do 7.5.


and kind of hit a wall right away.


That's why I said I'm


I need to practice.


Most of this stuff should be obvious, but


OK, so


can you, um, do I need to zoom in or anything? I don't think I can zoom in comfortably.


and very tight


There we go


All right. So do we need any


well, let's just go through the theory.


It always helps


yes


Hopefully it won't confuse us


So recall that a rational fraction is function as a fraction, function is defined as a ratio of 2 polynomials.


in the form of PPX divided by Q x.


or P and Q are polynomials in X and PFU obviously can't, can't be have a root, can't have a root.


that will um


affect the expression.


Actually, that's not right. I can't say it can't have fruits it's just that it, we don't consider


the times when uh P(x) has the root. We just don't include that in our domain.


cause we know that they'll be undefined.


Um, and this is assuming you can't just


factor out and then get rid of the term.


It just won't be, it just be irrational at that point.


So you can factor out and get rid of the roots.


oh okay


Anyway.


Um, so if the degree of PF X


is less than q of x.


the rational fraction is called proper.


Right, because just like we have our um


fractions by itself, when we have a fraction that is, that has a numerator that is smaller than the denominator, we call that a proper fraction.


right? Because you can't divide anymore.


right


Whereas if the numerator is greater than the the denominator, then you have an improper fraction because then you can


pull it out like, you know, 4/3, something like that, and you can, it's like 1 and 1/3, um


you get the point


So the improper, so it's otherwise, OK, the improper or rational facts could be reduced to the proper rational right.


Function by long division process. Thus, if POX is improper, then P over Q is equal tot of X + R(x) over Q of X.


which is um realizing now important.


I was trying to do a problem and it gave me this and I


feel like I applied it wrong.


TFX is a polynomial in X and


R(x) over Qx is a proper rational fraction.


as we know how to integrate polynomials, the integration of any rational function is reduced to the integration of a proper rational function.


So basically, if you get a


um


rational function


and you find that the degree of the enumerator is greater than the degree of the denominator


Then you perform long division.


to get a form.


um to get a uh


the function into a form of a proper function.


right? So that PE is always equal to or less than q. The the degree, I'm sorry.


I think specifically it has to be


it doesn't say what happens when the degrees are equal to each other.


We're just assuming from the table


that


wait, can i get 5 mins pls


the degree of the numerator


yeah


take your time


Mhm


i'm back


Welcome back.


thank you!


Hm


let's continue


Yeah, um.


so


what um


so I said, I, I went right where we left off. So thus, if P over q is improper, i.e.


um


if


oh, OK, if, if the degree of P


is greater than or equal to


the degree of Q


then their function is called proper.


and if it's improper


then we need to um


do long division to make sure that P over q is improper, which means that


even if P has the same degree as Q.


you have to perform long you have to perform long division to get


this form


to be able to make it um


integradable, eat more or, or be able to redefine it such that it's easier to integrate.


convert into mixed fraction?


kind of


Yeah, you want to convert it into mixed fraction, basically.


okay


Yeah


Um


yeah, it's, and then, uh, let's see. And R(x) is a polynomial inex.


P Q y over Q is a proper rational function, as we know how to integrate polynomials, integration of inter rational function is reduced to the integration of a proper rational function.


So they're saying that integration of any rational function.


can be reduced to the innovation of a proper rational function or saying more than likely that they're saying that you pretty much have to do it.


The rational fractions, which


we shall consider here for integration purposes will be those whose denominators can be factored into linear and quadratic factors.


So it's saying right here that the ones that they will consider, means that there are other forms with higher degree denominators, but they're just only considering linear and 1st and 2nd order.


denominators


ohh okay


And then assuming we want to evaluate where P overq is a proper rational function it's always possible to write the innergra.


as the sum of simpler rational functions by using partial fraction decomposition, which is basically what you're saying.


Um, kind of


you know, if it's greater, then you have to kind of mixed fractions, um, but


just trying to write it as something that's more understandable.


Then after this, the integration could be carried out easily using the already known methods, and that's the


big thing here, the past few chapters, it's just they've been giving you known methods.


right? All these different tools


so if you can figure out how


the, the square peg can fit this can fit the whole.


then, you know


it certainly helps to


uh solve the problem.


Cause I mean, you know, when you look at a problem like, oh, if I can just rewrite this to be like a square hole, then I can use the square peg.


or something else


right


Um, and then the following table 7.2 indicates a type of simpl fractions that are being associated.


associated with various kinds of rational fractions.


All right, so here


we have


um this first stipulation threw me off for a I I rethink about it. I don't know why, but


give me one sec


sure


Right, I think I just assumed it said A cannot equal zero, and I was throwing me off.


Um, but this first one


this saying that A cannot equal zero,


it's kind of an important s uh stipulation.


because if, if A equals 0, then we'd just be left with x^2 - a^2 minus x minus x1, which is already given in the previous chapter.


ohh right, we have directly formula for it


So they're just, yeah.


right now they, they're just kind of


stipulating that they have to be, just so that we have, we already, we already solved the other one we care about when these are the other cases of um


the denominator being different of the denominator being different, right? You already solved x^2 - a^2 minus a. Now we care about when it's


X minus A and X + X minus B. And one thing you notice, um


is, there is no coefficient for X, it's just 1.


All the everything the denominator has no eco efficient, which means that whatever form you get, you have to factor out


the coefficients


right


um and pull them out of the inner grand. I mean, you don't have to pull them out of the innergra, but you do have to pull them out


um,


of the expression


gotcha!


cause it may be useful to pull, to pull it out of the denometer and then you kind of


can reduce the numerator, but sometimes it won't happen, so you may just have to pull it out of the inner grand entirely.


So one thing that's interesting is


this um


squared term in the bottom x minus a square.


where it actually comes out to these two forms here.


the kind of X over A over X minus A and then big B over X minus B / (x-a)^2.


That's interesting


yeah, they took it like that so that lcm can be (x-a)^a


Yeah


yeah


um


he's got multiple, yeah. Um


another thing here is that, you know, using, oh jeez, what? I don't want to talk to co-pilot. Jeez, go away.


Um


sorry, I'm just being harassed by


Microsoft, uh


and one thing, you know, using this method, you will


get um


A, B, or C, you will get fractions.


because you will have, because you do have to pull out the coefficients for X, you will more than likely have fractions like X minus 1/2, you know, X minus 2/3, you know, so on and so forth. So just


kind of be mindful of that.


Then


yeah, right


here's another one


right.


LCM here.


and just added a separate term.


but this is a third-order cubic function in the bottom.


and same here.


It says we're Ax^2 + BX + C cannot be factorized further.


Right? Cause if you did, then you just have


um 3 then.


OK, that's the theory.


Now for example.


Um


I kind of want to screenshot this.


Do you know how to just take a pic, uh, take a screenshot of the tab


I forget how to do that


yeah


you can take snip


Oh yeah, snippy snip


it's win + shift + s


When shift


S


Hm


Actually, you know what? I wanna take a snippy snip of the previous ones too.


Oh, why did they separate these? Why couldn't they just have a nice table?


Girl?


Gir.


Oh, wind shift S, sorry you had to be here to watch this.


Just some housekeeping.


Um.


not this.


Just snipping crazy.


you can zoom out a bit, to add that one in


Um, I, are there any other forms?


Well, I guess I could have


Mm, lean out.


Hit that page divider


I'm sorry, hate is a strong word. I dislike it.


Yeah


Um, let's see. Are there any other forms


that I think it's


True identities


you have to save this snips too, if wanna see them later


cause I'm too lazy


Oh shoot, did I not save them?


No, I think they're saved.


or they're


screenshot copied the clipboard.


automatically saved to screenshots folder.


trying to scare me


I'm on to you


they are save to clipboard


not folder


They said, well, my tab said they're safe to automatically save to screenshots folder.


So I would have to


let me verify, is it Windows E? Yeah.


Um, search, uh, screenshots.


No, yeah, they're saved to my screenshots folder.


just to prove a point.


Yeah


is


here's image.


there


Can you see it


what?


I put it on the board, one of the screenshots.


ohh, i see it.


And then I can toss this other one up there too.


Boom


i have to save them myself


No, I feel like I could have just used the other screenshots, this page divide is annoying.


Do you really


yeah


Um, if you want, let me


if you want to be, if you want to be able to


screenshot it yourself.


We can just make this window


so no, I guess it doesn't work. Never mind.


You have to screenshot of a screenshot, I guess.


Can you actually save


the


if I upload something, can you actually save it?


I mean, I guess you could save the image.


no


No, I never thought of that


You can't save it


question mark


no ican't


Mhm. Um.


let me, can you take a snip of


the screen share


yourself


yes


I mean, actually you have the textbooks


might be more


easier for you then they just go into the textbook and scre and snip it.


I keep forgetting you're the source.


yeah, i can do that too


good point


done!


All right


You're gonna start remembering these pages, this page 45, um, this page 47, Pre-Calculus. I think I prefer the


document page rather than the textbook page.


cause you can't really navigate using a textbook page. Oh, stop. Co-pilot, stop harassing me. How do I turn you off?


I want to figure it out in my own time how to turn off co-pilot because this is just harassment.


OK, well


let's do some examples


So, feel free to go ahead and tackle


for example 11


okay


I have problem 6 is just beating me up.


Sorry, I scrolled down, let me.


put it back


done


All right, let's see. Did you get


a log of X + 1 over X + 2.


I mean, I got natural log.


yeah


I keep forgetting I like to use log.


They like to use log. That's annoying.


Um, I don't know what they mean by log. I don't think they're saying log base 10 or log base E, I guess it's logbase 10, but I don't think


i used ln too


um hold on one sec. What's the derivative of log space 10.


Yeah, 1 over X natural log of 10. Yes, so what it is natural log. I don't know why they keep using log


dummies


dumb, dumb dummies.


All right.


We can move on to example 12.


Now we have


um


this is different than what we did last exercise


what is different


Sorry, I talked to myself too fast and I forgot what point we were


there is no differentiation for denominator


and for what point you're trying to, there's no differentiation for denominator.


That is because there is no coefficient.


for X


They do have a differentiate for denominator, but the coefficient is what? Oh, I see what you're saying.


No, it's still the same thing cause the coefficient of


of actually of the coefficient is still one. So technically you do have differentiation, but it's just one, so it doesn't matter.


But they didn't feel the need to ride it out.


to write it out, right.


Cause we can go back


we have here


if we do, 1 = a [(x+1) (x+2)] + b, will we get same thing?


right there


I think we will


Um, actually, I think you would need more


I mean, I think we will get the same thing just because, you know, we're still talking about, for example, an apple


right, we're still talking about the same thing. We're just kind of trying to rewrite it differently.


but I think we'd still end up with the same result.


Like if you were to


divide it all out, you'd still get the same thing.


Um


one equals A, let me go back.


I'm jumping too much


Too many hoops. There we go.


So you said


1 equals


A X + 1, X + 2 + B. So you're saying, what if we didn't separate the denominator?


Actually, let's do it your way. Do it on the board.


Let's see if you can prove that they're the same.


When in doubt, do it out


okay


You have the board


You can't see me, but I'm flourishing my hand as a showman.


Did you get 2 x + 3?


Where did you get it


Uh, yes, yes, yes, yes, yes, yes, I see it.


Continue, please


Wait, wait, wait. Wouldn't it be the other way around? Wouldn't it be 3X?


Ah no, you're right. I'm sorry, I'm sorry. I'm way off.


No, you can be 01


not 0


it became problem


Right


So now you see why they did it the way they did.


because really the only way to do it


right


Or, I mean, you would, you would need


more variables if you're doing it your way.


But you have too many equations.


I'm sorry, too little equations


but it is what it is. So, it's good exercise.


b needs a equation


yeah, understood


I mean, technically, you have what, 12,


you have 3 equations, technically.


cause you have a


you have the X square term, you have the X term, and then you have the X to the 0 term. So you have 3 equations technically. So you could, you could have up to 3.


um, variables.


All right, that was fun


Now we can do


um


an improper fraction.


So example 12 is


an improper fraction


Oh, I'm sorry, can you see the screen?


Actually, are you ready to move on? Do you have any more questions?


um, about why they're not using the previous chapter's methodology. Actually, let me go back.


We we were talking about it so much. Why don't they have DX in there?


Um


um, so if you go back


or if you're looking at the screen, you see in the integral


integral DX over AX2 + BX + C, and you see that you have all these coefficients, right?


But what's the first thing they do?


right


is they factor out A


and you know, why is this important?


Well


if they didn't factor out A


and you have this um


AX


you try to rewrite, and you try to rewrite these expressions, you'd have, like, as you have here, um


yeah, if you have A piece and you try to


uh factor it


you would still end up with this kind of A over B and, and sorry B over A and C over A, um.


then when you do your substitution for tea.


right? And you had this AX term here.


then your


you, then you'd end up with A times DX equals dT. So you don't have to worry about this 1/8 term again and again.


which would, which would result in this


but they just find it easier to have


um DX equals DT straight up, but you already have this 1/8 term that they factored out. So it, it's, it'd be the same.


They just um find it easier to do it.


this way


or they just prefer, I'm not sorry, not easy, they prefer to do it this way.


So they're already accounting for this one over a term.


yeah, right


Yeah, and the method they used here was completing the squares.


not like um


the chapter we're on, which they don't use completing the squares, they prefer


having two separate terms, two separate expressions.


right? X minus A, X minus B.


So that's another difference


this also kinda easy too


I mean


don't speak too soon


All right. You haven't, we haven't really done many problems.


But the methodology, I guess, is simpler. You don't have to carry about the


um substitution method


yeah,


So that is, but that is a little easier, but you do have to do more, um, yeah, I guess you don't have to think as much.


But you do have to do more um


computations


right


You know, I'm trying to think of


what would you say, knowing what you know now, what would you say


is the difference or hm how do I say this?


or how do I ask this


No, I guess, I just be kind of, what kind of summary would you say between these two methods? Like what's the purpose of substitution method.


Um, what's the purpose of


the partial fraction decomposition.


you know, why are um


why do we need them


What's the


the purpose of the methodology.


I know it's kind of a loaded question.


but I just kind of want you thinking like


you know, what's, you know, why are we even bothering?


for every method we discussed


And why are we using these methods versus some other


imaginary method that we haven't thought of.


Maybe not for every method we discuss, just kind of in general.


I just wanted to kind of have you conceptualize and crystallize


your thought processes regarding these um


methodologies


so that you know kind of like, not only when to use them, but why.


and which method


you know, over another, cause you may have


problems that um


kind of overlap


and you can use one over the other.


And it, it might come down to preference, but you could also have um


you know, you want the solution of a certain form.


So you would use one over the other and just kind of like when, like why and when would you do that?


I know this is a, this is a lot to ask. Um, if you don't feel comfortable answering, just, just kind of be like uh.


and just say, you know, because they tell me to, but I just wanted you to um


kind of think critically


about these methods


and, you know, the rationale behind them.


okay, lemme think


Of course, take all the time you need


it depends upon the question. there can be multiple way to solve a prob, i'll go for the shorter one


True.


It's very pragmatic


yeah, i need a quick way for exam.


True


Right, for your needs, yeah.


OK. Well, we got that out the way.


So we can move on to


for example 12


Unless you have more to say


that was it? i thought it was like a conceptual like "why would you" question


I mean


it was


but then I realized that you're um


you're more only doing this because you have to.


Um


but it was like


I was thinking more along the lines of like


substitution method.


is


a way of


defining what we don't know.


right


Where


if we see something, if we can redefine it.


right, the in a grant, the, the integral, if we can re redefine it as something we know.


or an expression we know, then this allows us to


work with, work with it


because we, we've turned, we've turned unknowns into knowns.


right? And it's essentially like


dissecting it, dissecting the problem by looking at it, by kind of forcing it through a different lens.


and this allows us to work with it.


um,


where previously it was unworkable.


right? But the reason why we're using substitution method is because the, we can't integrate normally.


We take in something that's kind of unworkable, and we've turned it into something


um


just by looking at it differently.


rewriting the expression, right, the expression itself hasn't the problem itself hasn't changed. All we're doing is changing how we're looking at it.


right? We're redefining the problem.


and working it and working it that way.


versus um


you know, partial fracture decomposition.


where


you have this expression that you know all about


thanks to all the work we did and uh pre-calc, you know, that there's asintose, there's this, there's that.


Um


but then, you know, how does that relate to integration, it's like how do you find


Now, what does the area under the curve mean and all that. Um, we'll say, well, we can


redefine this


If we can divide it all out.


right? We've taken this


proper function. We've taken this function and turned it into like a


a proper function


right, which seems kind of silly that we, we've done all this in like algebra.


and done this sort of


silly methodology


and we get something that's actually workable.


Um, surprisingly, almost.


and come out with these, you know,


typically we're going to get log results, natural log results.


um, which, you know, isn't, it comes out kind of clean sometimes and then you, you're dealing with libra properties. But that's kind of like


you know, it's, it's interesting


how we, how you can do that cause technically you can


go use, if we went back to um


the rig identities


where are they? Technically we could use substitution method in partial fraction decomposition.


um if there's a denominator. Um


and solve it that way, you know.


And it's not, it's another thing that you can only use


partial fashion decomposition when you have um


A denominator that is not that simple.


numerator


right? The deno where the denominator is not just a constant. You actually have an expression as your denominator.


Oh man, I got all these eraser crumbs on me.


The numerator


Well, I mean, partial fraction and composition is when you have


an


expression in the denominator that is non-constant.


right? We actually have


X terms or cosine terms that just that uh polynomial of the degree than one.


which you technically do for


trig functions as well. Say you have tangent or something.


but that's not


a whole expression.


Where is the trigger identities?


I am just


I'm lost


at sea. Yeah, I think we're passed it.


I think it was 7.3.


Yeah, I think it was 7.3 immigration by substitution. Oh, substitution.


uh


true identities


after 7.3 i guess


Right, cause you could, you could solve this.


Um, it is 7.3.2, right? Cause there's problems you can solve just using substitution, right? Same thing here, substitution, um, kind of substitution.


actually won't be as clean.


But they're all just


ways of


defining the undefined.


which actually let me not use undefined because that


assumes that


in math terms, that's like infinity, you're dividing by zero or something. It's, it's a way of kind of grasping at the unknown.


right? We're just like, we have this object, um, it looks like other objects.


um, and we can make it look like other objects if we kind of


work it, if we look at it a certain way or we handle it a certain way or like a Rubik's cube you can kind of play with it a certain way.


Um, and then using


these tactics


substitution method, um, you know, trig methods, trick identities.


partial fraction decomposition.


then we can


solve them. They're basically algorithms.


Actually, they are algorithms. They're just methods.


right


right? Um


OK. Well, that's


are you, do you have any more questions on that topic or anything you wanna


consider


before we move on


Sorry if I got all of a sudden, so


not philosophical, but


no, i don't any questions right now.


It was mathematically conceptual.


i guess, idk


I mean, think about it. What is, what, what is an integral?


um


kind of crudely, it's like the area under the curve.


right.


Um, but what does that mean?


It's


I like to think of it as um integration as increasing dimensions.


where if you have


a length and you integrate it


it becomes an area. If you have an area, you integrate it, it becomes volume.


Um


and then


I don't know what the fourth dimensional form of volume is


the same thing for like acceleration, the velocity you're kind of adding these


dimensions


and for derivatives, it's the opposite. You're kind of taking away dimensions with like what if you consider it with respect to


time and then space and


you know, how do you


use these


their tools, but you're just kind of, you're, you're kind of changing


time and space as you use them, because really all you have to integrate with this time and space, or you could have units, you could have um


different, different things, but now you're adding these


um characteristics, sort of kind of getting meta with it when you have these real world applications.


and these, these methodologies substitution and everything.


are


kind of


keys to unlocking these dimensions.


I think I can end on that. That was cool.


i saw 4d in dr strange


Did you really


The movie or


I think, was there a show for Doctor Strange?


movie


Invented to Cumbretch is a movie actor, not a


marvel


TV. Well, I guess he did do, yeah, yeah, he did do a show.


um Sherlock


which I highly recommend. Great show.


with Sherlock with candidate Benedict was great.


sherlock homles?


though strap in because


it's they


each episode is like an hour and a half. Each episode is a movie in itself.


but it is good.


Yes, Sherlock Holmes


I think it's just called Sherlock, the show.


It was on Netflix, I don't know where it is now.


originally streamed on Netflix


ohh, i'll check it out. i had a book of sherlock homles


Did you really


With Moriarty and


I can't there he had other nemesis


the villains, but I think Moriarty was like his main nemesis.


And then you have the offshoots with um


Eurus, the sister


and the brother, they actually have the brother in the show.


the older brother


Well they have a tumultuous relationship.


i tried to read, couldn't finish it


I'm pretty sure the


girl from Stranger Things did Enola.


Yeah, I


I'm imagine that the tough read.


though I'm sure it's engaging once you get


to a certain point


That's how I found Lord of the Rings. It was, man, it was a tough to read, but then once you get past a certain point, it just, oh.


gets exciting


ohh lord rings, is hard to watch. i can't imagine read it. but that's great, you did it


Was it really hard to watch what's so difficult about it.


Um


it was 4hrs long!!


Yes, yes, it is.


I mean, it's more like 3 something, but yes.


That's because they've condensed so much.


your


chronicling


a massive story arc like that it's an era, you're chronicling.


I mean, imagine


trying to condense an era into


10 hours


A grand adventure.


with kind of a massive cast.


I mean, imagine if Star Wars was a book.


was a series of books, right?


be massive


I mean, Silmarillion itself, there's a lot of


right, i don't mind watching lond movie series. but i don't have much interst in history


examples that the books can go into


long**


uh history, which is more um


you know, speaking of books, it's more the lore.


The lore is kind of the history and the world building.


which kind of, which gives it substance.


and gives meaning


right? Now, instead of watching just a character


do something


you have more of the background of the character and now what they're doing has more meaning.


because now you have this, instead of a character, just kind of picking up a flower and then just tossing it aside. Now you have this forlorn character who's


been beriddled by loss and


detachment and more loss cause you just can't seem to stop attaching themselves to others and searching for


companionship in life and


losing it because either he's bad luck or he just


keeps attaching to characters that are soon to be lost.


and finds this flower


that reminds him of his mother.


and to see it tossed aside.


as a way of forgetting his own history.


Right now you've added gravitas to the scene, just by adding this lore, his background story


Anyway, that's


the arts


yeah, that kind of history is interesting.


But Lord of the Rings has a lot of um


a lot of lore. It has its own collection of books that are just


giving