SW1 FACTORING QUADRATIC EQUATIONS
This eighth grade mathematics lesson focuses on factorization of quadratic equations. It is the third lesson on factorization and the first one in which this technique is used for quadratic equations. The lesson is taught in Swiss German and is 47 minutes in duration. There are 24 students in the class.
| Time | Caption |
|---|---|
| 00:04:45 | Okay... Good morning... |
| 00:04:51 | We have been dealing with the subject of factorizing since the autumn holidays, which actually means the past two lessons. |
| 00:05:01 | And we have been working on two levels, on the one hand we have tried to find out |
| 00:05:09 | why factorizing actually is important, and in relation to what we saw yesterday, |
| 00:05:21 | that factorizing is important when calculating with fractions. |
| 00:05:27 | Because the subject is somewhat dry, we cheered ourselves up with these figures. |
| 00:05:35 | Today we will get acquainted with a second area where factorizing also is of great help, |
| 00:05:45 | and that is quadratic equations. Until now we were not able to solve quadratic equations. |
| 00:05:55 | With the help of factorizing, we will see that we are at least able to already solve some of them. |
| 00:06:04 | Last Tuesday we also did some exercises on how to deal with this factorizing technically. |
| 00:06:14 | And you solved problems for homework on this topic, which I now firstly would like to correct. |
| 00:06:29 | We have not discussed problem A yet, but that one doesn't pose any specific difficulties. |
| 00:06:42 | Maybe I'll repeat the method of proceeding right away: First we look at which parentheses, |
| 00:06:51 | which expression can be found in both addends. Here it is clearly parenthesis A minus B, which I will factorize- |
| 00:07:02 | meaning I will think about what this parenthesis multiplies with. The answer is: |
| 00:07:08 | It multiplies with, on the one hand, the three X plus four Y, and on the other hand, with plus two X plus Y. |
| 00:07:18 | We calculate it and we'll then notice, watch out, that there we have another common factor, which we can exclude again. |
| 00:07:27 | Are there any questions on this problem, on this result? |
| 00:07:43 | Now nobody is posing any questions. The second I don't have to discuss; we actually had solved that one |
| 00:07:53 | together as an example. That way we can proceed to the third problem. The method is exactly the same. |
| 00:08:07 | The bracket appears in both terms- I factorize them- what does this bracket multiply? |
| 00:08:19 | Ten Y plus seven A X minus this second bracket. |
| 00:08:25 | And now we have to summarize it, calculate it, and detect that still something more can be factorized here. |
| 00:08:43 | Is everything clear? Or nothing at all? Raphaela? |
| 00:08:48 | No. |
| 00:08:49 | What is it that isn't clear? Try to express your problem. |
| 00:08:54 | Well, there is a minus there... |
| 00:08:55 | This one? |
| 00:08:57 | Yes. |
| 00:08:58 | Yes. Well that is there. |
| 00:09:00 | [Laughter] |
| 00:09:02 | Now what about that minus sign? |
| 00:09:05 | Well, doesn't it mean something? |
| 00:09:08 | Yes, of course, minus. |
| 00:09:10 | [Laughter] |
| 00:09:15 | I have to multiply the difference of the two brackets with my bracket two M minus three N. |
| 00:09:25 | That's why I have to bring the minus over here. And your question is very good, because it refers to a difficulty: |
| 00:09:36 | We have to pay attention to that we write this bracket in any case. Because it is the whole expression that is subtracted. |
| 00:09:52 | The bracket must be there. This bracket here is actually optional. But this one I have to put, |
| 00:10:00 | because the minus works for both expressions. That's where the plus two B Y emerge from and they lead me to these twelve B Y. |
| 00:10:16 | Has your question been answered or not quite yet? |
| 00:10:24 | Well, one could also just calculate without the bracket. |
| 00:10:25 | Tomi. |
| 00:10:27 | Well- just go ahead and calculate instead of (inaudible) do the intermediate step. |
| 00:10:31 | Eh- just calculate it. How do you mean that? |
| 00:10:36 | Well simply calculate minus three A X plus two B Y now, and then leave the brack- bracket. |
| 00:10:41 | Sure if you're highly gifted, then you can do that. |
| 00:10:46 | There is just a high danger that you'll forget this combined sign change, |
| 00:10:55 | if you don't write the bracket. And you would easily write a minus here. |
| 00:11:04 | But it is certain that you could do it in one step. From this line you can conclude directly to this line here. |
| 00:11:12 | Now I solved the last two somewhat faster too, you'll see it in a minute... Are there any other questions about letter C? |
| 00:11:25 | Okay, the minus was the new difficulty in it, actually. |
| 00:11:31 | Now let's go to letter D. |
| 00:11:37 | Can you tell me what the common thing is? Which bracket can be found in both terms? Samuel? |
| 00:11:45 | Three U plus four V. |
| 00:11:49 | Three U plus four V, correct. I will factorize them here... And now I used Tomis' quick method... |
| 00:12:01 | Who can reconstruct how I got to these two terms, in this bracket? Yes, Samuel. |
| 00:12:13 | Six X Y plus four X Y and minus seven A- A B plus three A B. |
| 00:12:21 | Correct, equals minus four A B. I have factorized further because the |
| 00:12:28 | common factor two is contained in the number 10 and in the number four... And now there is nothing else that can be done. |
| 00:12:45 | E. |
| 00:12:51 | That has been written down in fast modus as well. I have left out the square brackets, |
| 00:12:58 | because that was originally only an aid to distinguish the brackets. |
| 00:13:05 | What is the common expression that I will factorize, Philipp. |
| 00:13:11 | NTwo A minus five B. |
| 00:13:12 | Two A minus five B... correct. Now how does the content of the second bracket develop, Philipp? |
| 00:13:22 | Seven M minus five N and minus three N mi- minus two N. |
| 00:13:27 | Good. Minus two N, hm. All right, that's the important minus again, which refers to the whole bracket here. |
| 00:13:42 | Minus five N and minus two N and now here there are no common factors, there is nothing more to do. |
| 00:13:53 | And the last one... F... Equals, Benjamin, what do we get? |
| 00:14:04 | NHm... two X squared minus one times nine Y. |
| 00:14:08 | Times nine Y, yes, you didn't specify, but one could hear from your speech that it stands in brackets, hm. |
| 00:14:18 | That was the common factor in the first sum. I take it out of the brackets and observe the plus sign. |
| 00:14:27 | Eight Y plus Y equals nine Y, Z plus minus Z cancels out, thus this will remain. You can see the difference. |
| 00:14:41 | At the beginning I wrote them out in detail, here I didn't do that anymore. I would advise you to write it... |
| 00:14:52 | relatively fast, meaning with fewer lines, but with great thought and exactness for what you are writing. |
| 00:15:01 | If you choose the detailed writing like here... the danger of copying is not far away. Then it could quickly happen that you will make a copying mistake. |
| 00:15:13 | I know these things. You might forget the two or you might make an X out of the Y and such things. |
| 00:15:25 | Are there any further questions concerning the homework? |
| 00:15:31 | Nothing else? Good. |
| 00:15:36 | Then we will continue to the second point... to these quadratic equations, which we want to take a closer look at. |
| 00:15:47 | And I'm now going distribute a worksheet to go with it. |
| 00:15:54 | The questions can be read on the worksheet. Give it some thought. First of all try to figure out in your notebook, |
| 00:16:06 | how one could do this. |
| 00:16:09 | You are very welcome to discuss- the matter- in pairs. We will fill in the correct solutions on the worksheet afterwards. |
| 00:16:37 | All right... |
| 00:16:54 | Here you've got these wonderful examples... |
| 00:17:02 | Have fun... |
| 00:17:08 | And also... |
| 00:17:09 | Thanks. |
| 00:17:10 | You're welcome. |
| 00:17:33 | (inaudible) |
| 00:17:34 | M-hm, yes. |
| 00:17:40 | Do you have to indicate the factorizing at the rear? |
| 00:17:43 | No, that's a transformation, the two terms are absolutely equal. |
| 00:17:49 | Minus in brackets. |
| 00:17:51 | Hm, what? |
| 00:17:52 | Minus in brackets. (inaudible) |
| 00:17:56 | Corine, I will correct it for you by tomorrow, all right? I don't have it yet. |
| 00:18:00 | M-hm, yes. |
| 00:18:09 | Do it in your notebook, Shane, so that you're able to make a clean entry of the solution here afterwards. |
| 00:18:18 | Yes and now- now you have to think about it... what does it mean... that is zero. Under what circumstances can- |
| 00:18:27 | Actually it should be a zero there, too... |
| 00:18:30 | Well, then write it down... From- one of the two (inaudible), that's a special case. |
| 00:18:36 | Ma'am, is that right (inaudible)? |
| 00:18:40 | I don't know if that's a beneficial beginning. |
| 00:18:41 | Can't you do that... |
| 00:18:44 | Do it first in your notebook. |
| 00:18:46 | Ye-es. |
| 00:18:50 | Hm... then it gets problematic- right, because you've got X squared and X. |
| 00:18:56 | How con you solve this? |
| 00:18:58 | Take a good look at- the term and maybe think a little about our "factorizing forceps"- |
| 00:19:09 | Or how did we call it- yesterday... there, our- our friend and helper. |
| 00:19:19 | Hm... Tomi! |
| 00:19:23 | Look, hm? |
| 00:19:24 | We met this one yesterday, right? |
| 00:19:28 | If we pose this question in connection with factorizing... maybe factorizing could help us along the track? |
| 00:19:43 | Can this be right? |
| 00:19:44 | That can very possibly be right. |
| 00:19:47 | (inaudible)... what I have to do. |
| 00:19:48 | That's right, now you have to think further. |
| 00:19:53 | What does this expression mean... and when can it be zero? |
| 00:20:00 | (inaudible) |
| 00:20:04 | No, that's not right. Five cannot be zero, Barbara. |
| 00:20:14 | Is it correct now like that? |
| 00:20:16 | Hmm... well, it is correct, but how do you continue now? |
| 00:20:24 | Well that's actually what I was going to ask about. [Laughter] |
| 00:20:25 | [Laughter] I- |
| 00:20:27 | Can't one the X over there- well- minus 25 on this side, |
| 00:20:32 | then it's minus 25 over there, then there is still that bracket over there... and then divide by this bracket on this side. |
| 00:20:38 | But you have got the X twice in there. |
| 00:20:42 | Yes, that's just it. |
| 00:20:43 | Well then. |
| 00:20:44 | If you calculate that, it becomes X to the second power again. |
| 00:20:45 | Did you really factorize? Look Raphaela, you still have a sum. |
| 00:20:52 | That's just it. |
| 00:20:53 | Isn't there a possibility to write this whole sum as a product? |
| 00:21:01 | Well, two. |
| 00:21:03 | That's right, yes- well... I think lots of you have at least done the first step... |
| 00:21:10 | of what does this X squared plus 10 X plus 25 remind you? Esti. |
| 00:21:19 | Of a binomial formula. |
| 00:21:21 | Of a binomial formula, correct- and which one... of them? What is it, this expression? |
| 00:21:34 | What, which- which case? |
| 00:21:35 | Yes. |
| 00:21:38 | The third? |
| 00:21:39 | Mh, not sure. |
| 00:21:41 | I don't know the number, I mean- |
| 00:21:42 | Yes, that doesn't matter. How can it be factorized, that was my original question... Michelle? |
| 00:21:49 | X plus five to the power of two. |
| 00:21:51 | X to the power of two... do the check. X squared plus two times the product- well 10 X, plus 25 is the square of five. |
| 00:22:05 | Is the square of five. This square is zero- how do we continue... Anette? |
| 00:22:16 | If one number times a number is zero, or a term times a term is zero, then one or both of them has to be zero. |
| 00:22:23 | And as both are the same, X is plus five equals zero. |
| 00:22:26 | X plus five has to be zero, correct- I've got that here... so that we don't have to write that much... correct. |
| 00:22:34 | You remember- we have seen this more than once. |
| 00:22:38 | If the product of two numbers or of two brackets is zero, one or the other bracket has to be zero. |
| 00:22:51 | And here I've got a special case. The A and the B are both X plus five. |
| 00:23:00 | This square can only be zero if X plus five is zero, which means that X has to have what value, Klaus? |
| 00:23:08 | Hm, does X have to be minus five? |
| 00:23:10 | X has to be minus five. |
| 00:23:14 | Okay we made it, to solve our first quadratic equation, sort of. |
| 00:23:24 | But you can see, it is a special case, we had a binomial formula. |
| 00:23:32 | Now you can chew the cud and think on- the second tricky problem... it is not much more difficult than the first one. |
| 00:23:44 | Try to find a solution for it and always keep the- law about products in mind. |
| 00:23:57 | Let's give everybody some time. |
| 00:23:59 | Please transfer the correct solutions from here to the theory sheet, which will then be complete. |
| 00:24:12 | Okay well, you can do a little mental calculation, right? |
| 00:24:15 | You know, yesterday I calculated (inaudible) 11 times 11 as basic area. |
| 00:24:21 | Eleven times 11 equals- |
| 00:24:23 | One hundred and twenty-one. |
| 00:24:24 | One hundred and twenty-one... yes... that's what it comes to. |
| 00:24:39 | What else is there, Sandra? |
| 00:24:45 | Eleven times what equals 121? |
| 00:24:49 | Eleven. |
| 00:24:54 | M-mh, yes. |
| 00:24:55 | Is this correct like that? |
| 00:24:57 | Yes that's correct. |
| 00:25:01 | Well. |
| 00:25:07 | Ho-ho, they are- |
| 00:25:08 | (inaudible) something like that... eh, and then... minus 14... 14. Is that correct? |
| 00:25:19 | There is just one trifle missing, it has to be equal. |
| 00:25:23 | Oh, and then it's correct? |
| 00:25:24 | That's true, yes... well done. |
| 00:25:26 | Is X 11? |
| 00:25:27 | X is 11 in the second one, look, I covered it at the front. |
| 00:25:30 | Ah. |
| 00:25:34 | Okay, you are so quick. Some of you have already solved the third one... Shane, move on. |
| 00:25:48 | That's the third one, it says A plus B and A minus B. |
| 00:25:50 | M-mh, yes. |
| 00:25:53 | Ehm... is X. |
| 00:25:57 | Our dear binomial formulas. |
| 00:26:04 | One hundred and nine- sixty-nine. |
| 00:26:08 | One hundred and sixty-nine. |
| 00:26:11 | Well... well, well... What is 12 times 12? |
| 00:26:22 | Look, you only have to look at the last number- nine. As a square you will only get that with a three. |
| 00:26:29 | Then it is quite close, that you... what have I got, one hundred- |
| 00:26:35 | Ah- excuse me, I read 169 from here, it is 196. |
| 00:26:41 | Ah, 196? |
| 00:26:42 | Right? No... I think... yes, it's 96. |
| 00:26:49 | Have you got the third one? |
| 00:26:51 | Eh, just until here. |
| 00:26:53 | All right, now there's a small difference. Well you can see- 196- |
| 00:27:03 | Yes, what did I write there, oh gosh. Probably I wanted to do 169. No, that's wrong. |
| 00:27:11 | The reflection stays the same, but the factorization is X minus 14 times X plus 14 equals zero. |
| 00:27:22 | I really did write 96 on the sheet, didn't I? Quickly take a look. Yes, yes. Very well. |
| 00:27:29 | And now it's a bit different than before. In the first two cases I had... |
| 00:27:37 | twice the same bracket as factors. And now I have two brackets. All right... let's think about this- rule here. |
| 00:27:49 | One of them is zero, or the other, thus X minus 14 must be zero, or- second variant- Benjamin. |
| 00:27:59 | X plus 14 has to be zero. |
| 00:28:01 | X plus 14 has to be zero, and that will give us for X the two solutions... |
| 00:28:08 | Fourteen or minus 14. |
| 00:28:11 | Fourteen or X equals minus 14. |
| 00:28:17 | Hansueli asked a question: can one write it down as a solution set, and if so, how. Barbara? |
| 00:28:27 | Well, eh, solution set equals, ehm, minus 14, and then well- ehm- yes... |
| 00:28:32 | Semi- semicolon. |
| 00:28:34 | Yes, semicolon and then 14. |
| 00:28:36 | And 14. I will list them, the two solutions. |
| 00:28:39 | You can't really see it anymore. |
| 00:28:40 | Anette? |
| 00:28:41 | Now could one just write absolute solution value 14? |
| 00:28:46 | Yes, you could. But you don't really do that. If you have an absolute value in the end, you split it up in the two numbers. |
| 00:28:59 | You see, we have thus found out that we now are capable of solving quadratic equations if we- |
| 00:29:09 | either recognize a first, a second or a third binomial formula. |
| 00:29:21 | It's just that quadratic equations are not always of this kind. And the next one is a rather tricky problem. |
| 00:29:32 | Now let's see, if you can find anything here... The idea is again that we try to find two brackets, whose product is zero, |
| 00:29:46 | and whose product corresponds to this expression six X squared minus seven X minus three. |
| 00:29:56 | What do you have to fill in the brackets, you can guess a little, try it out, and test. |
| 00:30:06 | Pardon? |
| 00:30:07 | Could one... a little there... |
| 00:30:09 | Yes, you won't need the above anymore. Thanks. I just don't want to show the solution yet. |
| 00:30:18 | Could you show that in the back there once again? |
| 00:30:25 | Like that! |
| 00:30:38 | Okay... good. Bravo. |
| 00:30:44 | Yes, now you're- I think it is correct- wait, I have to read it from your side... three equals minus nine, yes that's correct. |
| 00:30:55 | Now you have to finish the calculation, Marc. |
| 00:30:57 | Then X- equals- minus three point three. |
| 00:31:01 | Yes, maybe you should write it as a fraction- why three point- |
| 00:31:11 | Yes exactly, one third, correct. |
| 00:31:18 | Search for yourself! Don't let yourself get influenced by your neighbor. |
| 00:31:28 | Look, you have to think. There has to be two brackets. |
| 00:31:31 | How can you proceed in order to get six X squared in the product. |
| 00:31:38 | Well... three and two, right? |
| 00:31:42 | Three X and two X. Also possible would be six X and one X. |
| 00:31:47 | There you somehow have to make a decision, you have to try it out. |
| 00:31:52 | M-hm, yes. |
| 00:31:53 | There is no method through which you can immediately say what's correct. In the last place- it has to be a three. |
| 00:32:03 | M-hm, yes. |
| 00:32:05 | That is minus three, to be more exact. |
| 00:32:13 | Two X- yes- no, you've got a sign error there, watch out... If you have the number of three- here, then you take two X to the right. |
| 00:32:27 | Minus two X. |
| 00:32:28 | Minus or just two X equals plus three. |
| 00:32:32 | Well then ask her. |
| 00:32:35 | That would be one variant, yes. |
| 00:32:38 | (inaudible) six X and there X and three X and two X, two X are three. |
| 00:32:42 | Correct, okay the last variant, Shane, those are actually the same ones. You just wrote it the other way around. |
| 00:32:46 | Well yes, but- but- well- somehow it has to be the other way around... can it partially be too. |
| 00:32:52 | Yes, yes, all right you can test them both with that. Basically the two possibilities stand open. |
| 00:32:58 | Yes. |
| 00:32:59 | Correct and now you have to consider what has to come in the last place. |
| 00:33:02 | And to find what is correct, you ascertain that using the middle term, afterwards. |
| 00:33:08 | M-hm, yes. |
| 00:33:13 | M-hm, yes. |
| 00:33:20 | What have we got here? Yes in a minute... six X squared minus seven X plus three equals zero. |
| 00:33:39 | Now could you please look up for a second? The ideas that you came up with- for filling in the bracket- were- |
| 00:33:49 | Shane just mentioned it before, six X and one X, then I get the six squared. |
| 00:33:57 | Or, another possibility, which we actually should recognize quickly- is this variant. |
| 00:34:08 | We have done with the term- we needed it to get this far. Secondly, you were supposed to- |
| 00:34:19 | look at the number of minus three. How can you get the product minus three, with integers, for now. |
| 00:34:29 | Plus three times minus one. |
| 00:34:31 | Plus three times minus one. Now I could- put- plus three and minus one here, but I could also put minus one here, |
| 00:34:43 | and plus three there, and here I have also got the same options to choose from. |
| 00:34:51 | Well at the moment, I have different choices of an ending. What fits and- let's proceed this way- what does certainly not fit. |
| 00:35:06 | How can I recognize if one of these four solutions is the correct one... Samuel. |
| 00:35:15 | Well, it- I have to obtain seven X. |
| 00:35:19 | Good. The last one- the last term here in the middle, we haven't considered that one at all yet. |
| 00:35:29 | If I calculate these products I will get this term, right? |
| 00:35:35 | That's why we now have to think about which one of these four cases really gives us the minus seven X. |
| 00:35:42 | How about the first case? How many X do I get there? Minus six plus three- equals, Hansueli. |
| 00:35:58 | Minus three X. |
| 00:35:59 | Minus three X, all right... we can forget that, that's wrong. Second case... How many X do I get here? Jeanine. |
| 00:36:17 | M-mh- (inaudible) X. |
| 00:36:21 | I can't really follow you. Six times three X, six X times three? |
| 00:36:26 | Eighteen. |
| 00:36:27 | Eighteen. |
| 00:36:29 | And... |
| 00:36:30 | Minus o- well, one. |
| 00:36:31 | Minus one, 17 X... that's something else yet. The first case is dead. Maybe we'll have more luck with this? |
| 00:36:46 | There are... Corinne? |
| 00:36:50 | Nine X. |
| 00:36:51 | Nine X, and what else? |
| 00:36:54 | Minus one X. |
| 00:36:57 | Ehm, Excuse me, I showed you the wrong one, hm? Three X times minus one is? |
| 00:37:04 | Two X. |
| 00:37:08 | No. Three X times minus one is minus three X and then? |
| 00:37:17 | Plus five X. |
| 00:37:20 | Pay attention... don't be nervous. |
| 00:37:23 | Six X. |
| 00:37:24 | Six X, six minus three, three X is left in the middle... Our hopes are diminishing, if it's not going to work out this time, |
| 00:37:35 | the whole thing does not look very hopeful. Let's just write it down now. |
| 00:37:43 | Our last hope... is that correct? Nine X... and now, Marc? |
| 00:38:00 | He needs to put his glasses on. Nine X? |
| 00:38:04 | Ehm, nine X, minus- eh, two X equals seven X. |
| 00:38:08 | Equals seven X. Hm. |
| 00:38:13 | Now I've got six X squared plus seven X minus three. |
| 00:38:16 | Two (inaudible). |
| 00:38:22 | Thanks. Hm. Almost. Are we able to wangle that one? Hansueli. |
| 00:38:31 | Ehm, three X plus one and the second bracket, two X minus three. |
| 00:38:42 | Correct. Let's do the check. Is it really correct now? Yes, (inaudible)? |
| 00:38:48 | Let's multiply everything properly once again. |
| 00:38:53 | Ehm, six X squared. |
| 00:38:55 | Six X squared. |
| 00:38:58 | Ehm, minus nine X. |
| 00:39:00 | Minus nine X. |
| 00:39:02 | Two X. |
| 00:39:03 | Plus, minus? |
| 00:39:05 | Eh, plus. |
| 00:39:06 | Plus two X and? |
| 00:39:08 | Minus three. |
| 00:39:09 | Minus three equals zero, that's correct now... wonderful. Uf! Hm? |
| 00:39:24 | Well you can see... it doesn't always go that easy as we have seen with the first two problems that we did. |
| 00:39:36 | Tomorrow we will see how one can proceed to be able to solve something like that systematically. |
| 00:39:47 | But for today you have got enough new material. Now I would like you to practice a little on these techniques, because you will notice, |
| 00:40:00 | that you have to get some routine to solve these factorizations quickly. In your book you have... |
| 00:40:12 | Yeah, yeah, well, on the- worksheets to be more exact. Third grade book, well, on your copies, the numbers 14 A, |
| 00:40:22 | 15 B, and then also 17 B, they are all problems on this topic. It is written down, Benjamin, which problems. |
| 00:40:43 | Fourteen A clearly says: determine the solution for the equation. And you have multiple quadratic equations. |
| 00:40:53 | You have to try to solve them now, using this method. |
| 00:40:58 | (inaudible) |
| 00:41:00 | Yes, yes, sure, they should be entered on the theory sheet. Shall I put it up for you to see once more? |
| 00:41:05 | Yes, could you (inaudible). |
| 00:41:08 | Yes, please. |
| 00:41:09 | My pleasure... just one after the other, I'll write it down afterwards, okay? |
| 00:41:14 | There- I think you can read the last one, too, on the blackboard. |
| 00:41:24 | We know, right, our blackboard isn't that great, but it doesn't matter, we can still manage. |
| 00:41:33 | Should anybody still have questions, then raise your hand now so that I can pass by. |
| 00:41:37 | What problems do we have to do now, 14? |
| 00:41:40 | Look, it is written right at the bottom there, 14 A, 15 B, and 17 B. They are all equations that you have to solve. |
| 00:41:54 | Well the (inaudible). |
| 00:41:55 | The... sixt- 15 and 14 A, yes. |
| 00:42:04 | The solutions are there in the front as usual, you can check if your results are correct or not. |
| 00:42:12 | Eh, you're standing right in my way. |
| 00:42:14 | Yes. |
| 00:42:26 | Do I have to (inaudible) number 11 (inaudible) now? |
| 00:42:30 | What do you mean? |
| 00:42:32 | Ehm, now... I have. |
| 00:42:34 | Here we said number 14 A... Yes, yes, we will look at them too, for practicing. |
| 00:42:41 | But for what we are trying out right now- |
| 00:42:44 | Yes, eh. |
| 00:42:45 | It's 14A, 15B, and 17B, actually 16 too. |
| 00:42:53 | M-hm. Yes. |
| 00:43:01 | Does it work out? Homework was all right? |
| 00:43:06 | Hmm... yes, I continued a little bit there. |
| 00:43:09 | Yes, yes, yes, that's okay. |
| 00:43:10 | When you have solved some already, Christoph, you can just as well go and check your answers in the solution booklet if they are correct. |
| 00:43:16 | M-hm, yes. |
| 00:43:17 | It's laying in the front. |
| 00:43:19 | M-hm, yes. |
| 00:43:30 | M-hm, well it is actually all about recognizing the binomial formulas. |
| 00:43:35 | Yes, yes. (inaudible) then, eh, yes (inaudible) X minus three... |
| 00:43:40 | Yes, exactly. Shane, try to improve your presentation a little, will you? |
| 00:43:46 | It looks so very loose and careless [in French], how you're doi- Just look at how much space you're using! |
| 00:43:52 | You could use much less if you consistently wrote on every second line. |
| 00:43:58 | You know, when we are dealing with those fractions, as we saw in our first example there, and you're using that much space, |
| 00:44:05 | then you will need a whole page only for one problem and won't be able to keep the overview anymore, or everything will become too big. |
| 00:44:11 | Yes, sure. |
| 00:44:13 | Well, I would try to do that a little bit smaller. |
| 00:44:19 | Yes, good. |
| 00:44:22 | Ma'am. |
| 00:44:23 | That's correct. |
| 00:44:24 | Is that right? |
| 00:44:26 | Yes. |
| 00:44:27 | See. |
| 00:44:28 | Mrs. (inaudible), could you please come? |
| 00:44:30 | It's just a little bit strange that you always take your X to the right when you write it. |
| 00:44:33 | Yes, I did it differently this time. |
| 00:44:34 | It doesn't matter, it's just a bit uncommon. You can be original if you like. |
| 00:44:37 | I always do it like that. |
| 00:44:40 | (inaudible) 15 B, right? |
| 00:44:42 | Yes. |
| 00:44:43 | Yes, then this should be that or that, and there are these two solutions, right? |
| 00:44:47 | Yes- that's correct. |
| 00:44:50 | But Hansueli has a completely different solu... |
| 00:44:52 | For 15 B? No, he is working on 14. |
| 00:44:56 | Ah, yes, we just had to do the three listed in the front, right? |
| 00:45:00 | Yes you are doing problem 15 B and he is working on problem 14 A. |
| 00:45:07 | No wonder you have got different results. |
| 00:45:10 | And look, if you want to write it in short form, you can of course just list it after your consideration... |
| 00:45:21 | M-hm, yes. |
| 00:45:22 | All right, that has the same value. |
| 00:45:25 | Tomi, nice presentation, please. |
| 00:45:28 | Yes, I know. |
| 00:45:29 | Well, why don't you do it then? |
| 00:45:31 | Yes... |
| 00:45:32 | I don't know. |
| 00:45:33 | M-hm, yes. |
| 00:45:42 | Are you sure that it is minus, Jeanine? |
| 00:45:47 | No, plus. |
| 00:45:48 | M-hm, yes. |
| 00:45:51 | Yes, exactly, m-hm. You always have to pay attention there, those are the- the tricky errors that annoy you afterwards. |
| 00:46:03 | Yes, yes, you see, 15 B can hardly be considered as a B problem, because everything is so wonderfully factorized already, hm. |
| 00:46:16 | Ehm, Ma-m. |
| 00:46:17 | Five sixth, yes, correct. |
| 00:46:19 | Should I write seven minus X? You don't know where the minus could be. |
| 00:46:23 | But you must have X squared in front. Which one are you doing there? A. |
| 00:46:28 | That one, D. |
| 00:46:29 | Eh, D. It has to be X squared. |
| 00:46:33 | But (inaudible). |
| 00:46:35 | Aha, it cou... Yes, yes, sure. If you change both the signs then it stays the same. |
| 00:46:40 | If X minus seven is zero, then X is seven, and if seven minus X is zero, then X is also seven. |
| 00:46:48 | M-hm, yes. |
| 00:46:49 | See, it doesn't matter in which order you write it. |
| 00:46:55 | Ehm, does it matter which one I do first? |
| 00:46:58 | Eh, the negative or the positive. Now do you mean at the listing? |
| 00:47:00 | Yes. |
| 00:47:02 | No. Mostly you state the solution set in enumerating form, beginning with the smallest value. Do you know- |
| 00:47:10 | Then it's still correct like that? |
| 00:47:11 | It is also correct. One can imagine it directly on the number line that way. |
| 00:47:17 | Yes. |
| 00:47:18 | That's the advantage, hm, but... |
| 00:47:24 | How do I have to do this now? (inaudible) |
| 00:47:28 | If it was binomial... then it would be a first one, right? Well, an X plus something squared. |
| 00:47:36 | One hundred and forty-four is the square of what number? |
| 00:47:42 | Of 24. |
| 00:47:43 | Hm-m, no. |
| 00:47:44 | No. |
| 00:47:48 | Ten times 10 equals one hundred. Eleven times 11 we saw before, and what's next? |
| 00:47:55 | Twelve times 12. |
| 00:47:56 | Equals just about 144. And now look, if you take X plus 12 in the middle, |
| 00:48:03 | you will, as is generally known, receive the double product. And two times 12- correct, equals 24. |
| 00:48:13 | Now if you had 27 here, then you wouldn't be able to factorize it like that, hm. |
| 00:48:21 | So you mustn't deceive yourself and think that it will always work like that. These are of course problems that all work. |
| 00:48:29 | But in other cases it could look completely different. |
| 00:48:36 | M-hm, yes, you can also just write it as a square, Esti, you don't have to write them twice next to each other. That's not necessary. |
| 00:48:41 | Oh yes. |
| 00:48:43 | That's not necessary. |
| 00:48:46 | There is no square number here in number E. |
| 00:48:50 | Where? |
| 00:48:51 | E. |
| 00:48:52 | E. |
| 00:48:53 | E... Two X and three in the back... doesn't that work out? |
| 00:49:01 | Ah, two X times- eh, plus three. |
| 00:49:04 | Squared, right, because there's a minus here and maybe you have to do something with the signs, too. |
| 00:49:12 | All right, the bell is going to ring in a minute, or maybe it already has. For tomorrow, please just do this first problem, 14 A. |
| 00:49:23 | The whole thing? |
| 00:49:24 | All of it. It goes fast. |
| 00:49:26 | No! |
| 00:49:27 | You will need a little routine. |
| 00:49:33 | Did you solve the wrong one? |
| 00:49:35 | No, I did two F and the next two. |
| 00:49:36 | Yes, well then there's not much left. All right. We will continue with this tomorrow. |
| 00:49:45 | Oh, you poor thing. What else? |
| 00:49:52 | Is that right in the brackets there? Because that is- they skipped it just like that. It is not finished yet, I know, but (inaudible). |
| 00:50:00 | But now there is no A plus B anymore there in the back. |
| 00:50:04 | Goodbye everyone! |
| 00:50:05 | Goodbye to you all! |
| 00:50:07 | You factorized A plus B here. |
| 00:50:11 | M-hm, yes. |
| 00:50:12 | Nine X plus eight X minus that, but it is- that's that. |
| 00:50:14 | Is that minu- is that minus or |
| 00:50:17 | That's that. |
| 00:50:18 | (inaudible) that it becomes a minus. |
| 00:50:19 | Right. |
| 00:50:20 | Yes I just wanted- well- |