WEBVTT 00:00:00.940 --> 00:00:06.506 This video is about how to solve quadratic equations. Let's begin 00:00:06.506 --> 00:00:11.566 by saying exactly what a quadratic equation is. It's an 00:00:11.566 --> 00:00:17.638 equation that must contain a term that has got X squared in 00:00:17.638 --> 00:00:24.216 it, so we can have three X squared minus five X squared, or 00:00:24.216 --> 00:00:30.288 perhaps just X squared on its own. The equation can also have 00:00:30.288 --> 00:00:38.150 terms in. X so it might have a term 5X or perhaps minus 7X 00:00:38.150 --> 00:00:44.420 or perhaps North Point 5X. It can also have constant terms 00:00:44.420 --> 00:00:47.270 just numbers, so perhaps 6. 00:00:47.790 --> 00:00:51.780 Minus Seven a half, etc. 00:00:52.810 --> 00:00:57.217 It can't have any other terms in it. It can't have any higher 00:00:57.217 --> 00:01:02.302 powers of X in it, so it can't have X cubed in it. Can't have 00:01:02.302 --> 00:01:04.675 things like one over X in it. 00:01:05.200 --> 00:01:10.429 So what does our most general quadratic look like? 00:01:10.430 --> 00:01:16.820 AX squared it must have this X squared term in it and a is just 00:01:16.820 --> 00:01:23.210 a constant. It can be one as it is here, One X squared. It could 00:01:23.210 --> 00:01:28.748 be 3. It could be minus five. It could be any real number. 00:01:29.370 --> 00:01:35.769 Plus BX, now this term may not be there. 00:01:36.270 --> 00:01:38.478 Be could be 0. 00:01:39.600 --> 00:01:45.144 Plus, CC is the constant term and again that term doesn't have 00:01:45.144 --> 00:01:48.378 to be there. See could be 0. 00:01:49.090 --> 00:01:55.360 And then we say equals Nord, so that's our most general 00:01:55.360 --> 00:02:00.438 quadratic. And what we're going to have a look at is how to 00:02:00.438 --> 00:02:03.174 solve all the different varieties of this kind of 00:02:03.174 --> 00:02:04.694 equation that you can meet. 00:02:05.660 --> 00:02:09.386 There are four 00:02:09.386 --> 00:02:15.990 basic ways. 1st Way is by Factorizing. 00:02:16.540 --> 00:02:22.156 I'm going to assume that you know how to factorise, but there 00:02:22.156 --> 00:02:27.772 is another section of the video that is to do with Factorizing 00:02:27.772 --> 00:02:32.452 quadratic expressions. So if you're not sure how to factorise 00:02:32.452 --> 00:02:37.600 perhaps it would be a good idea to look at that. 00:02:39.130 --> 00:02:43.090 2nd method is by 00:02:43.090 --> 00:02:47.645 completing. The 00:02:47.645 --> 00:02:53.460 square. Now again, I'm going to assume that you 00:02:53.460 --> 00:02:57.600 know how to complete the square, but again, that is another video 00:02:57.600 --> 00:03:01.395 that relates to completing the square, so if you're not sure 00:03:01.395 --> 00:03:05.535 how to do that, perhaps you might look at that video first. 00:03:06.590 --> 00:03:12.778 Then we're going to have a look at how you use the formula. 00:03:13.420 --> 00:03:16.036 There's a special formula for solving quadratic equations. 00:03:16.036 --> 00:03:17.998 We're going to be using that. 00:03:18.650 --> 00:03:24.220 And finally, how to solve a quadratic equation using graphs? 00:03:25.140 --> 00:03:30.756 Of these four methods, two of them are by far the most common. 00:03:31.480 --> 00:03:35.290 And Whilst one doesn't want to say you mustn't understand 00:03:35.290 --> 00:03:40.243 everything, you can pick a part of what you do. You really do 00:03:40.243 --> 00:03:43.291 need to concentrate on factorising and the formula. 00:03:43.291 --> 00:03:45.577 Those are the two most important 00:03:45.577 --> 00:03:53.276 ones. So let's begin by having a look at how to factorise and 00:03:53.276 --> 00:03:55.508 solve a quadratic equation. 00:03:55.550 --> 00:04:02.072 I'll begin with three X squared 00:04:02.072 --> 00:04:04.246 equals 27. 00:04:05.640 --> 00:04:10.476 There is no extra, but that still doesn't stop it being a 00:04:10.476 --> 00:04:14.506 quadratic equation 'cause it's got this X squared term in. 00:04:15.110 --> 00:04:22.166 So we begin by writing it equals 0, so I take the 00:04:22.166 --> 00:04:24.518 27 of each side. 00:04:25.580 --> 00:04:32.030 So we have three X squared minus 27 equals 0. 00:04:32.790 --> 00:04:37.698 That's the first step, and one that you must do every time. 00:04:37.698 --> 00:04:43.015 Next, look for a common factor. Is there a common factor in the 00:04:43.015 --> 00:04:48.332 terms on the left hand side of this equation, and if there is, 00:04:48.332 --> 00:04:53.240 let's take that common factor out and here we have three X 00:04:53.240 --> 00:04:59.375 squared and also 27 which has a factor of three in it. 27 is 3 00:04:59.375 --> 00:05:03.874 times by 9, so we can take out the common factor. 00:05:03.900 --> 00:05:11.852 All three and that will leave us with X squared minus 9 equals 0. 00:05:13.060 --> 00:05:18.868 Inside this bracket, now we have the difference of two squares. 00:05:18.868 --> 00:05:25.732 We have X squared takeaway 9 which is 3 squared and this is 00:05:25.732 --> 00:05:31.012 a standard piece of Factorizing. So that's three brackets, X 00:05:31.012 --> 00:05:34.180 minus three X +3 equals 0. 00:05:35.270 --> 00:05:39.086 Now when we multiply 2 expressions together and we're 00:05:39.086 --> 00:05:44.598 not quite sure of their value, but we do know what they give 00:05:44.598 --> 00:05:49.262 us. In this case zero. We can make certain deductions about 00:05:49.262 --> 00:05:53.926 them, so if I have one number multiplied by another number 00:05:53.926 --> 00:05:58.166 and the answer is 0, then one of these numbers. 00:05:59.550 --> 00:06:06.830 That one or that one or both of them are zero. So we have 00:06:06.830 --> 00:06:09.430 X minus three equals 0. 00:06:10.060 --> 00:06:17.060 All X +3 equals 0, so we end up with 00:06:17.060 --> 00:06:22.660 X equals 3 or X equals minus three. 00:06:24.050 --> 00:06:31.310 Notice 2 answers and that will be a theme that I will 00:06:31.310 --> 00:06:33.125 return to again. 00:06:35.170 --> 00:06:36.832 Let's just recap there what we 00:06:36.832 --> 00:06:42.886 did. First of all, we put everything equal to 0. Then we 00:06:42.886 --> 00:06:48.012 extracted, took out any common factors. Why should you do that? 00:06:48.012 --> 00:06:52.672 Well, let's just run through this one. Again, 3X squared 00:06:52.672 --> 00:06:59.196 equals 27 and let me break a few of the rules that I've just 00:06:59.196 --> 00:07:04.322 said. So let me cancel both sides by three, since obviously 00:07:04.322 --> 00:07:08.516 three goes into three X squared and leaves us. 00:07:08.520 --> 00:07:12.810 X squared and free goes into 27 and leaves us with nine, so 00:07:12.810 --> 00:07:16.770 I've got X squared equals 9. Now let me do the natural 00:07:16.770 --> 00:07:20.400 thing which is to take the square root of both sides. 00:07:20.400 --> 00:07:25.350 Square root of X squared is X and the square root of 9 is 3. 00:07:27.410 --> 00:07:30.896 Only got 1 answer if we 00:07:30.896 --> 00:07:33.960 look back. 2 answers. 00:07:34.550 --> 00:07:41.180 I've lost one by factorizing. I was able to get two factors and 00:07:41.180 --> 00:07:43.730 that gave Me 2 answers. 00:07:44.500 --> 00:07:51.039 In doing this step from here to here and taking the square root, 00:07:51.039 --> 00:07:58.081 what I have forgotten is that as well as 3 squared be equal to 00:07:58.081 --> 00:08:05.123 9 - 3 squared is also equal to 9. So at this point I 00:08:05.123 --> 00:08:08.141 should have said or X equals 00:08:08.141 --> 00:08:13.494 minus three. Now it's very easy to make this kind of mistake and 00:08:13.494 --> 00:08:18.170 lose one of the answers. So the best way to tackle these is the 00:08:18.170 --> 00:08:20.174 way that I just set out. 00:08:21.040 --> 00:08:27.384 Write them as equals. 0 take out common factors and then look at 00:08:27.384 --> 00:08:31.288 a factorization leading to giving you 2 answers. 00:08:31.890 --> 00:08:38.046 Let's take another example. This one will take us five X squared 00:08:38.046 --> 00:08:44.202 plus three X equals 0. It's already written as equals 0, so 00:08:44.202 --> 00:08:50.358 we don't have to worry about that first step. So the second 00:08:50.358 --> 00:08:53.949 step is to look for a common 00:08:53.949 --> 00:09:01.360 factor. I've got five X squared plus 3X, so I've got 00:09:01.360 --> 00:09:08.764 a common factor here of X so I can take that out, 00:09:08.764 --> 00:09:12.466 leaving me with five X +3 00:09:12.466 --> 00:09:17.207 equals 0. And here again I've got 2 numbers. 00:09:17.960 --> 00:09:23.462 X and 5X plus three. I'm not sure what they are, but what I 00:09:23.462 --> 00:09:27.785 do know is that in multiplying them together, I've got 0. 00:09:28.330 --> 00:09:35.066 This means one of them must be 0 or the other one must be 0, or 00:09:35.066 --> 00:09:40.960 they're both 0, so I can write down X equals 0 or 5X. Plus 00:09:40.960 --> 00:09:46.433 three is also equal to 0, so there's one answer, X equals 0, 00:09:46.433 --> 00:09:50.643 and now this is just a simple linear equation, linear. 00:09:50.643 --> 00:09:56.116 Remember because it's just got an X in it. Now X squared's ex 00:09:56.116 --> 00:09:58.642 cubes, no one over X is. 00:09:58.660 --> 00:10:05.104 Just an X and this solves quite easily. 5X equals minus three, 00:10:05.104 --> 00:10:10.474 taking three away from both sides and then dividing both 00:10:10.474 --> 00:10:14.233 sides by 5 - 3 over 5. 00:10:14.890 --> 00:10:20.610 So there are my two solutions, X equals 0 and X equals minus 00:10:20.610 --> 00:10:26.330 three over 5. What I'd like to do is just perhaps run through 00:10:26.330 --> 00:10:31.610 this one again and show you where the mistakes can come when 00:10:31.610 --> 00:10:35.570 tackling equations like this. 'cause what's special about this 00:10:35.570 --> 00:10:41.290 one is there's no constant term an ex squared term, an ex term, 00:10:41.290 --> 00:10:43.050 but no constant term. 00:10:43.070 --> 00:10:46.400 So what might be the mistake that we would 00:10:46.400 --> 00:10:47.880 make when doing this? 00:10:49.790 --> 00:10:56.188 Let me state that we might very well make it say, Oh yes, a 00:10:56.188 --> 00:11:02.129 common factor of X one there in the X squared term and one 00:11:02.129 --> 00:11:08.070 there, so I'll cancel by that. X5 X +3 equals 0, dividing each 00:11:08.070 --> 00:11:14.925 term by X and then 5X is equal to minus three, and then X is 00:11:14.925 --> 00:11:17.667 equal to minus three over 5. 00:11:19.560 --> 00:11:23.460 One answer X equals minus 3/5. 00:11:23.960 --> 00:11:30.610 If we go back with X equals 0 and X equals minus 3/5, we've 00:11:30.610 --> 00:11:34.885 lost this answer. Here we've lost X equals 0. 00:11:35.620 --> 00:11:40.960 Why? Because we didn't look for a common factor at this point. 00:11:40.960 --> 00:11:45.855 We went straight on to simply divide throughout by the X 00:11:45.855 --> 00:11:51.640 instead of taking it out as a common factor. So you must be 00:11:51.640 --> 00:11:55.645 aware of that with quadratic expressions, were looking for 00:11:55.645 --> 00:12:00.985 two answers to solutions to roots. Those are if you like the 00:12:00.985 --> 00:12:04.990 different words that we use. Answers, solutions, roots, they 00:12:04.990 --> 00:12:06.778 all mean. The same thing. 00:12:07.290 --> 00:12:13.192 OK, we looked at two particular cases. One where we did not have 00:12:13.192 --> 00:12:18.640 an exterm and another one where we didn't have a constant term. 00:12:18.640 --> 00:12:21.870 So now. Let's have a look at. 00:12:22.650 --> 00:12:29.146 One that's got both of those in. 00:12:29.390 --> 00:12:33.020 X squared minus 5X plus 6 00:12:33.020 --> 00:12:38.067 equals 0. We're going to solve this by Factorizing. 00:12:38.710 --> 00:12:43.738 So to factorize a quadratic we need to search for two numbers 00:12:43.738 --> 00:12:47.928 that when we multiply them together will give us this 00:12:47.928 --> 00:12:50.460 number here. 6. 00:12:51.200 --> 00:12:57.380 An when we add them together will give us this number here 00:12:57.380 --> 00:13:05.062 minus 5. So we look at that and 3 * 2 would give us 6 and 00:13:05.062 --> 00:13:09.605 minus 3 N minus two added together would give us minus 00:13:09.605 --> 00:13:14.561 five, but minus three times minus two is also six, so those 00:13:14.561 --> 00:13:19.104 look like good choices, minus three times by minus two gives 00:13:19.104 --> 00:13:24.060 us 6 and minus three plus minus two gives us minus 5. 00:13:24.660 --> 00:13:29.325 So. We take this expression again X squared. 00:13:30.210 --> 00:13:36.788 Instead of minus 5X, we write this as minus three X minus two 00:13:36.788 --> 00:13:43.872 X +6 equals 0. Now we look at these two terms and we take 00:13:43.872 --> 00:13:50.450 out the common factor. In this case that's X and that leaves us 00:13:50.450 --> 00:13:53.486 with X Times X minus three. 00:13:54.030 --> 00:13:59.840 Now we look at these two terms and we take out again a common 00:13:59.840 --> 00:14:04.820 factor. Well obviously there's two, but this is minus 2X, so it 00:14:04.820 --> 00:14:10.630 cost. This is minus two X I'm going to take, minus two as the 00:14:10.630 --> 00:14:16.025 factor minus two brackets X. And now I need to put something in 00:14:16.025 --> 00:14:21.420 here so that I have minus two times gives me plus six, and 00:14:21.420 --> 00:14:23.495 that must be minus three. 00:14:24.030 --> 00:14:30.166 Equals 0. Now I've got one lump of algebra there. An 00:14:30.166 --> 00:14:35.134 one lump of algebra there, and they share this common Factor X 00:14:35.134 --> 00:14:40.516 minus three X minus three. So we take that out as a common 00:14:40.516 --> 00:14:47.690 factor. I'm we're left with the other factor as X 00:14:47.690 --> 00:14:54.590 minus two X minus two. Close the bracket equals 0. 00:14:55.110 --> 00:14:59.994 And so here I've arrived at 2 numbers multiplied together and 00:14:59.994 --> 00:15:07.098 they give me 0. So one of them must be 0 or the other one must 00:15:07.098 --> 00:15:10.206 be 0 or they perhaps both 0. 00:15:10.570 --> 00:15:17.374 So if we just write that down again, X minus three times by X 00:15:17.374 --> 00:15:19.318 minus 2 equals 0. 00:15:20.620 --> 00:15:27.232 Then either X minus three equals 0 or X minus 2 equals 00:15:27.232 --> 00:15:34.395 0. This tells us that X must be equal to three, or this 00:15:34.395 --> 00:15:39.905 one tells us that X must be equal to two. 00:15:40.990 --> 00:15:46.372 And so again, we've arrived at two possible answers. 00:15:46.910 --> 00:15:50.840 Let's take another one. This time, let's take something that 00:15:50.840 --> 00:15:55.949 does not have a unit coefficient in front of the X squared, so 00:15:55.949 --> 00:15:57.914 will take 2 X squared. 00:15:58.490 --> 00:16:01.794 Plus 3X minus 2 00:16:01.794 --> 00:16:08.326 equals 0. Now, in order to solve this one, we're looking 00:16:08.326 --> 00:16:13.100 for two numbers that will multiply together to give us 2 00:16:13.100 --> 00:16:17.874 times by minus two SO2 numbers that multiply together to give 00:16:17.874 --> 00:16:23.082 us minus four where looking for those same 2 numbers to add 00:16:23.082 --> 00:16:25.252 together to give us 3. 00:16:26.350 --> 00:16:33.252 Well. Four times by one would give us four and if we 00:16:33.252 --> 00:16:38.868 made the one A minus 1 four times by minus one, that would 00:16:38.868 --> 00:16:44.052 make minus four and four plus minus one would give us 3. 00:16:44.820 --> 00:16:50.969 So these are our two numbers that we need, so again 2 X 00:16:50.969 --> 00:16:56.172 squared and then the X term we can rewrite us 4X. 00:16:56.970 --> 00:17:04.061 Minus X. Minus 2 equals 00:17:04.061 --> 00:17:10.532 0. And we look at these two terms for a common factor, 00:17:10.532 --> 00:17:17.706 and there's a 2 here and four is 2 * 2 and there's an X and an 00:17:17.706 --> 00:17:23.614 X&X in the X squared and X with the 4X. So our common factor 00:17:23.614 --> 00:17:26.568 there is 2X, leaving us with X 00:17:26.568 --> 00:17:32.274 +2. And now we want to common factor here and that will be 00:17:32.274 --> 00:17:38.908 minus one. Times X +2 equals 00:17:38.908 --> 00:17:45.744 0. Again. We have two lumps of algebra 00:17:45.744 --> 00:17:52.240 here and here, and in each one there's a factor of X +2, so 00:17:52.240 --> 00:17:58.736 that's our common factor that we can take out X +2 times by two 00:17:58.736 --> 00:18:02.500 X. Minus one. 00:18:03.490 --> 00:18:08.060 So again, we've arrived at two brackets multiplied together X 00:18:08.060 --> 00:18:15.372 +2 times by two X minus one, so one of these has got to be 0, 00:18:15.372 --> 00:18:17.657 or possibly both of them. 00:18:18.550 --> 00:18:25.240 Let's write that down again. X plus 2 * 2 00:18:25.240 --> 00:18:31.261 X minus one equals 0. So either X +2. 00:18:31.920 --> 00:18:34.509 Is 0 or. 00:18:35.090 --> 00:18:38.870 2X minus one is 0. 00:18:39.530 --> 00:18:45.970 This tells us that X is minus 2. By taking two away from both 00:18:45.970 --> 00:18:52.410 sides. This tells us that 2X is equal to 1. By adding one to 00:18:52.410 --> 00:18:58.850 each side, and now we divide both sides by two. So we have X 00:18:58.850 --> 00:19:04.730 equals 1/2. So we ended up with our two answers for this 00:19:04.730 --> 00:19:07.974 quadratic equation. Let's take 00:19:07.974 --> 00:19:09.448 another one. 00:19:09.760 --> 00:19:14.360 4. X squared 00:19:15.380 --> 00:19:18.114 +9 equals 00:19:18.114 --> 00:19:23.900 12 X. It doesn't say equals 0. 00:19:24.400 --> 00:19:27.627 So let's write it so it says 00:19:27.627 --> 00:19:32.780 equals 0. Four X squared minus 12 00:19:32.780 --> 00:19:35.740 X +9 equals 0. 00:19:36.880 --> 00:19:39.600 Check common factor. Something we didn't do in the last 00:19:39.600 --> 00:19:42.864 question, but then there wasn't one. But these numbers are a bit 00:19:42.864 --> 00:19:46.400 bigger and there may just be a common factor there. But as we 00:19:46.400 --> 00:19:48.032 look there is no common factor. 00:19:48.710 --> 00:19:53.150 Factorise it, we're looking for two numbers that would multiply 00:19:53.150 --> 00:19:58.478 together to give us 36 four times by 9. Ann would add 00:19:58.478 --> 00:20:01.142 together to give us minus 12. 00:20:01.920 --> 00:20:06.008 Well, six and six give us 36 and if we make them both negative. 00:20:06.980 --> 00:20:14.414 Minus 6 times by minus six they still give us plus 36, but now 00:20:14.414 --> 00:20:21.317 they add together to give us minus 12. So we've got four X 00:20:21.317 --> 00:20:26.627 squared minus six X minus six X +9 equals 0. 00:20:27.340 --> 00:20:33.281 In these two terms, we want the common factor and that will be 00:20:33.281 --> 00:20:39.679 2X because this is four and six and X squared and X, so will 00:20:39.679 --> 00:20:45.620 take the two X out, giving us 2X minus three, and then here 00:20:45.620 --> 00:20:50.434 again. Common factor 6X9. There's a common factor. There 00:20:50.434 --> 00:20:55.998 are three, but this is a minus sign here, so we want minus 00:20:55.998 --> 00:21:01.990 three times by two X. That gives us the minus 6X and minus three 00:21:01.990 --> 00:21:07.982 times by something has to give us plus 9, so that will be minus 00:21:07.982 --> 00:21:09.694 three again equals 0. 00:21:10.260 --> 00:21:16.552 Two lumps of algebra. A common factor in each one of two X 00:21:16.552 --> 00:21:23.970 minus three. Leaving us with two X minus 3 * 00:21:23.970 --> 00:21:30.870 2 X and two X minus three times minus three. 00:21:32.050 --> 00:21:37.990 One of these two or both of them can be 0. 00:21:38.490 --> 00:21:45.390 So we have two X minus 3 * 2 X 00:21:45.390 --> 00:21:52.290 minus three equals 0 and so 2X minus three equals 00:21:52.290 --> 00:21:57.810 0 or two X minus three equals 0. 00:21:58.990 --> 00:22:04.730 Adding three to both sides 2X is equal to three, and so X is 00:22:04.730 --> 00:22:06.780 equal to three over 2. 00:22:08.310 --> 00:22:12.756 Of course, this is exactly the same, so we can write down the 00:22:12.756 --> 00:22:16.860 same stuff. Two X equals 3 and X equals 3 over 2. 00:22:18.340 --> 00:22:19.759 2 answers again. 00:22:20.580 --> 00:22:25.680 The same answer twice, because I can say it like that. It 00:22:25.680 --> 00:22:30.355 enables me to keep up this idea that quadratics are going 00:22:30.355 --> 00:22:35.880 to lead me to 2 answers and answer may be repeated, but I 00:22:35.880 --> 00:22:37.580 still get 2 answers. 00:22:38.810 --> 00:22:42.670 Let's take one more example. 00:22:44.640 --> 00:22:51.136 X squared minus three X minus 2 equals 00:22:51.136 --> 00:22:57.236 0. We're looking for two numbers that will multiply 00:22:57.236 --> 00:23:02.656 together to give us minus two and will add together 00:23:02.656 --> 00:23:05.366 to give us minus three. 00:23:06.720 --> 00:23:10.790 I gotta multiply together to give us minus two. Well, that 00:23:10.790 --> 00:23:15.970 would suggest one and minus two, but no matter how we try and add 00:23:15.970 --> 00:23:20.040 one and minus two together, we're not going to get minus 00:23:20.040 --> 00:23:24.850 three. And if we said well, let's try minus one and two and 00:23:24.850 --> 00:23:29.290 again and not going to add together to give us minus three. 00:23:29.910 --> 00:23:33.285 In short, this question does 00:23:33.285 --> 00:23:41.034 not factorize. And because it doesn't factorize, we're 00:23:41.034 --> 00:23:48.666 going to have to look for another way 00:23:48.666 --> 00:23:51.528 of solving it. 00:23:52.120 --> 00:23:57.528 So how are we going to solve an equation that does not factorize 00:23:57.528 --> 00:24:02.520 well? One option is completing the square. I want to have a 00:24:02.520 --> 00:24:07.512 look at this, not because it's the regular way in which we 00:24:07.512 --> 00:24:11.672 would try to solve an equation that doesn't factorize, but 00:24:11.672 --> 00:24:16.664 be'cause. In doing it, we'll see how the next method which is 00:24:16.664 --> 00:24:21.240 using the formula actually works. So let's have a look at 00:24:21.240 --> 00:24:22.488 what we've got. 00:24:22.530 --> 00:24:29.344 X squared Minus three X minus 2 equals 0. Now in order to 00:24:29.344 --> 00:24:35.376 complete the square, we look at these two terms. Now this is an 00:24:35.376 --> 00:24:42.336 X squared and this is minus 3X. So if I want to get a complete 00:24:42.336 --> 00:24:44.656 square which remember is like 00:24:44.656 --> 00:24:50.990 that. And I've got to have an X in there and I've got to have a 00:24:50.990 --> 00:24:55.410 constant term here so that when I do the squaring I'll end up 00:24:55.410 --> 00:24:56.430 with minus 3X. 00:24:57.080 --> 00:25:03.500 And we know how to do that. You take a half of the number that 00:25:03.500 --> 00:25:08.636 multiplies the X, so that's minus three over 2, so that when 00:25:08.636 --> 00:25:13.344 we multiply out, that bracket will get X squared, will get 00:25:13.344 --> 00:25:18.908 minus 3X, but will also have an extra term. We will have the 00:25:18.908 --> 00:25:21.476 square of minus three over 2 00:25:21.476 --> 00:25:27.247 extra. In other words, we'll have added on minus three over 2 00:25:27.247 --> 00:25:32.928 squared, so if we've added it on, we better take it off in 00:25:32.928 --> 00:25:38.172 order to make sure that we've got exactly the same value, the 00:25:38.172 --> 00:25:42.979 same expression, and then, of course, I've got minus two here, 00:25:42.979 --> 00:25:45.164 so let's put that in. 00:25:45.190 --> 00:25:51.010 So now this expression is exactly the same as that one, so 00:25:51.010 --> 00:25:57.315 let's keep this bit at the front X minus three over 2 four 00:25:57.315 --> 00:26:02.165 squared, and now what's this? Let's simplify this here. Well, 00:26:02.165 --> 00:26:05.560 minus three over 2 squared is 9 00:26:05.560 --> 00:26:09.166 over 4. Take away 2 00:26:09.166 --> 00:26:14.286 equals 0. So we've got X minus three 00:26:14.286 --> 00:26:16.350 over 2 all squared. 00:26:17.370 --> 00:26:20.786 Minus, now let's write this all over 4. 00:26:22.080 --> 00:26:27.096 Two over 4 is 8 quarters. I'm taking away nine quarters and 00:26:27.096 --> 00:26:30.858 I'm taking away eight quarters. Sign takeaway. 17 quarters 00:26:30.858 --> 00:26:35.038 altogether. Now this now looks more manageable. Let me just 00:26:35.038 --> 00:26:38.800 turn over the page and write this down again. 00:26:40.610 --> 00:26:47.522 X minus three over 2 all squared minus 00:26:47.522 --> 00:26:50.978 17 over 4 equals 00:26:50.978 --> 00:26:58.078 0. Now we're going to add the 17 over 4 to both sides, so will 00:26:58.078 --> 00:27:03.776 have X minus three over 2. All squared is equal to 17 over 4. 00:27:03.776 --> 00:27:08.660 And now I'm going to take the square root of both sides, 00:27:08.660 --> 00:27:12.730 remembering that I must get plus or minus this side. 00:27:13.310 --> 00:27:19.904 So the square root of this will be X minus three over 2 equals. 00:27:20.440 --> 00:27:25.900 To take the square root of this I need the square root of the 00:27:25.900 --> 00:27:31.360 top, which is going to be plus Route 17 over 2 or minus Route 00:27:31.360 --> 00:27:36.430 17 over 2 over the two, because that's the square root of the 00:27:36.430 --> 00:27:43.946 four. Now I add this bit to both sides to three over 2, 00:27:43.946 --> 00:27:51.478 so we have X equals 3 over 2 plus Route 17 over 2 or. 00:27:52.050 --> 00:27:59.630 X equals. The square root of 17 over 00:27:59.630 --> 00:28:05.540 2. And just for the sake of completeness, I'm going to 00:28:05.540 --> 00:28:11.870 put all of this over this too. So 3 Plus Route 17 over 2 or. 00:28:12.950 --> 00:28:16.250 3 minus Route 17 00:28:16.250 --> 00:28:22.250 over 2. So that we've done a question that involves 00:28:22.250 --> 00:28:23.675 completing the square. 00:28:24.220 --> 00:28:29.740 Again, we've got 2 answers, but there are two further things to 00:28:29.740 --> 00:28:34.340 notice here. I've left these square roots in. You would 00:28:34.340 --> 00:28:38.480 normally take a Calculator and workout an approximate value. 00:28:39.080 --> 00:28:44.468 Of these two answers to a given degree of accuracy, that might 00:28:44.468 --> 00:28:49.407 be to two decimal places, it might be to three significant 00:28:49.407 --> 00:28:53.897 figures, but these are exact answers. The minute you put 00:28:53.897 --> 00:28:57.938 decimals in those answers are not exact, they are 00:28:57.938 --> 00:29:00.183 approximations to a given degree 00:29:00.183 --> 00:29:06.114 of accuracy. The third thing I want you to notice is there form 00:29:06.114 --> 00:29:10.722 3 plus the square root 3 minus the square root and over 00:29:10.722 --> 00:29:14.946 something because when we look at the next method, you're going 00:29:14.946 --> 00:29:16.866 to see that form again. 00:29:17.590 --> 00:29:23.882 So how else might we tackle this equation to remember what it was 00:29:23.882 --> 00:29:29.690 initially? It was this One X squared minus three X minus 2 00:29:29.690 --> 00:29:36.466 equals 0. We saw that it didn't factor eyes, so we did it by 00:29:36.466 --> 00:29:38.886 this method of completing the 00:29:38.886 --> 00:29:46.091 square. So what we're going to do now is look at the 00:29:46.091 --> 00:29:53.423 formula. To do this, I'm going to take it generally to begin 00:29:53.423 --> 00:29:59.533 with. So here is our general quadratic equation AX squared 00:29:59.533 --> 00:30:03.199 plus BX plus C is 0. 00:30:03.840 --> 00:30:09.624 And there is a formula which will solve this equation for us. 00:30:10.390 --> 00:30:14.488 And it's X equals minus B. 00:30:15.810 --> 00:30:18.159 Plus or minus. 00:30:18.980 --> 00:30:26.804 And it's the plus or minus that's going to give us the 00:30:26.804 --> 00:30:33.976 two answers. The square root of B squared minus four AC 00:30:33.976 --> 00:30:39.192 all over all over whole lot over 2A. 00:30:40.600 --> 00:30:43.760 That's the formula for solving quadratic equations, and 00:30:43.760 --> 00:30:48.895 unfortunately you do have to learn it. But if you say it to 00:30:48.895 --> 00:30:54.425 yourself each time you write it down, X is minus B plus or minus 00:30:54.425 --> 00:30:59.560 the square root of be squared minus four AC all over 2A helps 00:30:59.560 --> 00:31:04.300 to remember. It makes it sound stupid, but it helps to remember 00:31:04.300 --> 00:31:09.435 it. So let's take this formula and use it to solve the previous 00:31:09.435 --> 00:31:11.410 equation that we just had. 00:31:11.460 --> 00:31:19.020 So the equation that we had was X squared minus three X 00:31:19.020 --> 00:31:25.950 minus 2 equals 0. Let's begin by identifying AB and see 00:31:25.950 --> 00:31:32.880 there's One X squared. So a must be equal to 1. 00:31:33.420 --> 00:31:41.328 B is the coefficient of X, and that's minus three, so B 00:31:41.328 --> 00:31:49.236 equals minus. 3C is the constant term, so here C is equal 00:31:49.236 --> 00:31:57.144 to minus two. Let's write down the Formula X equals minus B 00:31:57.144 --> 00:32:04.393 plus or minus square root of be squared minus four AC. 00:32:04.990 --> 00:32:08.842 All over 2A by writing down 00:32:08.842 --> 00:32:14.790 each time. It's a way of helping to remember it. Now let's 00:32:14.790 --> 00:32:16.434 substitute these values in. 00:32:17.110 --> 00:32:20.904 So B is minus three, so that's 00:32:20.904 --> 00:32:23.530 minus. Minus 00:32:23.530 --> 00:32:31.049 3. Plus or minus the square root of 00:32:31.049 --> 00:32:38.419 B squared, minus three all squared minus four times a, 00:32:38.419 --> 00:32:40.630 which is one. 00:32:40.720 --> 00:32:46.880 Times C, which is minus two, will just extend that square 00:32:46.880 --> 00:32:53.600 root sign all over everything over 2A, which is 2 times by 00:32:53.600 --> 00:33:00.415 one. Now let's tidy this up. So we have minus minus three. 00:33:00.415 --> 00:33:01.960 That's plus 3. 00:33:02.770 --> 00:33:05.896 Plus or minus the square root 00:33:05.896 --> 00:33:09.605 of. Minus 3 all squared is 00:33:09.605 --> 00:33:16.260 9. Four times by one is 4 times by minus two is minus 8 00:33:16.260 --> 00:33:20.560 and we've got this minus sign here, which makes +8. 00:33:21.570 --> 00:33:23.928 All over 2. 00:33:24.860 --> 00:33:28.340 And what we see we've got is 3. 00:33:29.210 --> 00:33:35.720 Plus or minus the square root of 17, all over two, which are the 00:33:35.720 --> 00:33:40.835 answers that we had before. Again, we can workout each of 00:33:40.835 --> 00:33:46.880 these answers. 3 Plus Route 17 over two or three minus Route 17 00:33:46.880 --> 00:33:51.995 over 2 using a Calculator. But again, remember any answers you 00:33:51.995 --> 00:33:56.645 get doing it that way. Our approximate. These answers are 00:33:56.645 --> 00:34:02.143 exact. Let's take one more example of using the formula. 00:34:02.920 --> 00:34:09.993 And the one that will take this time, three X squared 00:34:09.993 --> 00:34:12.565 equals 5X minus one. 00:34:14.010 --> 00:34:20.432 It doesn't say equals 0, so we must write it as equals 0, 00:34:20.432 --> 00:34:25.866 'cause That's the form that the formula demands. We have the 00:34:25.866 --> 00:34:31.794 equation written in, so three X squared and we gotta take 5X 00:34:31.794 --> 00:34:33.770 away from each side. 00:34:33.960 --> 00:34:39.784 And we've got to add 1 to each side. That ensures that we have 00:34:39.784 --> 00:34:45.940 that. Now let's identify AB&CA is the coefficient of X squared, 00:34:45.940 --> 00:34:49.012 which in this case is 3. 00:34:49.700 --> 00:34:57.008 B is the coefficient of X, which in this case is minus 00:34:57.008 --> 00:35:04.022 5. See is the constant term, which in this case 00:35:04.022 --> 00:35:11.676 is one. Again, let's write down the equation. X is minus 00:35:11.676 --> 00:35:15.480 B plus or minus the square 00:35:15.480 --> 00:35:22.074 root of. B squared minus for 00:35:22.074 --> 00:35:25.497 AC all over 00:35:25.497 --> 00:35:30.370 2A. And make the substitution. 00:35:31.390 --> 00:35:39.006 B is minus five we want minus B, so this is minus minus 5. 00:35:39.010 --> 00:35:42.229 Close all minus. 00:35:42.230 --> 00:35:49.000 Square root of B squared, so that's minus five all 00:35:49.000 --> 00:35:55.770 squared, minus four times a, which is 3 times C, 00:35:55.770 --> 00:36:02.540 which is one, and that's all over 2A, two times 00:36:02.540 --> 00:36:05.248 a, which is 3. 00:36:05.930 --> 00:36:12.680 Now simplify this minus minus five is +5 plus or 00:36:12.680 --> 00:36:19.430 minus the square root of minus five. All squared is 00:36:19.430 --> 00:36:25.708 25. 4 * 3 * 1 is 12, and the minus sign is minus 00:36:25.708 --> 00:36:29.138 12. All over 6. 00:36:30.300 --> 00:36:37.132 So we just squeeze that one into this space. Here we 5 plus or 00:36:37.132 --> 00:36:44.452 minus the square root of 25 - 12, which is 13 all over 6. And 00:36:44.452 --> 00:36:49.820 again there are two exact answers there. 5 Plus Route 13 00:36:49.820 --> 00:36:56.652 over 6 and 5 minus Route 13 over 6 we can workout their value 00:36:56.652 --> 00:37:00.556 approximate value that is by using a Calculator. 00:37:00.590 --> 00:37:05.621 And finding out what it tells us about Route 30, but those are 00:37:05.621 --> 00:37:07.943 two exact answers. If left in 00:37:07.943 --> 00:37:13.114 that form. Now basically, if a quadratic factorizes, then you 00:37:13.114 --> 00:37:16.618 should solve it by using factorization, 'cause it's 00:37:16.618 --> 00:37:18.808 clearly much quicker than this. 00:37:19.860 --> 00:37:22.737 Don't use completing the square unless you 00:37:22.737 --> 00:37:26.847 specifically asked to and use this method by using the 00:37:26.847 --> 00:37:30.546 formula. Solving by using the formula for those that 00:37:30.546 --> 00:37:31.779 do not factorize. 00:37:32.880 --> 00:37:37.584 However, right at the very beginning we did say there was a 00:37:37.584 --> 00:37:42.970 fourth method. And so I just want to look quickly at what 00:37:42.970 --> 00:37:44.486 this 4th method is. 00:37:46.110 --> 00:37:49.526 We said this 4th method was by graphing. 00:37:50.560 --> 00:37:56.240 OK. Let's have a look at what we know about quadratic 00:37:56.240 --> 00:37:59.579 functions. If X squared is positive in our quadratic 00:37:59.579 --> 00:38:04.031 function, then we're going to get a U shaped curve like that. 00:38:04.031 --> 00:38:08.854 If, on the other hand, X squared is negative, then we're going to 00:38:08.854 --> 00:38:10.709 get a Hill like that. 00:38:11.350 --> 00:38:14.470 Now. Set on a graph. 00:38:15.450 --> 00:38:17.148 Then we might have a graph. 00:38:17.710 --> 00:38:22.671 That looks with a positive X squared. Say something like that 00:38:22.671 --> 00:38:28.534 or we might have a graph that say looked like that we might 00:38:28.534 --> 00:38:31.240 have a graph that looked like 00:38:31.240 --> 00:38:35.176 that. Now this is the 00:38:35.176 --> 00:38:40.566 line. Equals 0 the value of the function equals 00:38:40.566 --> 00:38:46.650 0, and so here we can see our two values of X. 00:38:47.690 --> 00:38:53.566 Here we can see one value of X which makes the function equal 00:38:53.566 --> 00:39:00.208 to 0. And we've said we call that a repeated root or a double 00:39:00.208 --> 00:39:05.330 route, but here the graph does not reach the X axis, it does 00:39:05.330 --> 00:39:10.452 not reach a value equal to 0. And that means we actually have 00:39:10.452 --> 00:39:14.392 no roots. So there are some quadratic equations that you 00:39:14.392 --> 00:39:17.544 cannot actually solve well. Mathematicians make it happen, 00:39:17.544 --> 00:39:22.272 so to speak, they make them solvable by inventing a new kind 00:39:22.272 --> 00:39:27.394 of number. But we're not going to deal with that at the moment. 00:39:27.470 --> 00:39:31.265 As far as we are concerned, we either get 2 routes. 00:39:32.420 --> 00:39:33.740 A repeated root. 00:39:34.390 --> 00:39:35.710 Or for the moment. 00:39:36.230 --> 00:39:37.358 No roots at all. 00:39:38.180 --> 00:39:42.730 But you will see that we can make an equation like this have 00:39:42.730 --> 00:39:46.930 two routes in a very special way. Clearly what we've done for 00:39:46.930 --> 00:39:51.480 a positive X squared term we can do for a negative X squared 00:39:51.480 --> 00:39:56.030 term, because again we can have a graph say that looks like that 00:39:56.030 --> 00:40:00.930 and does not reach the X axis. Or we can have one that's a 00:40:00.930 --> 00:40:05.480 looks like that and just touches the X axis. Or we can have 00:40:05.480 --> 00:40:07.230 another one that looks like 00:40:07.230 --> 00:40:13.564 that. And again two values there and there a repeated value of X 00:40:13.564 --> 00:40:16.588 there and no values of X there. 00:40:17.120 --> 00:40:21.014 So bearing that in mind, let's have a look at our. 00:40:21.790 --> 00:40:27.718 Trial equation if you like the one that we've used to start off 00:40:27.718 --> 00:40:33.646 each of the last two sections, X squared minus three X minus 2 00:40:33.646 --> 00:40:40.030 equals 0, and let's think how we can solve this by using a graph. 00:40:40.620 --> 00:40:43.050 Let's just look at this bit. 00:40:43.590 --> 00:40:49.050 The X squared minus three X minus two and think of it as a 00:40:49.050 --> 00:40:53.340 function, not as an equation equals 0, but just a function. 00:40:53.340 --> 00:40:59.580 Now one of the things is that if we put in, say, a value of X 00:40:59.580 --> 00:41:03.870 equals what. Let's say something like one, then we've got one. 00:41:04.910 --> 00:41:10.760 Minus 3 - 2 is clearly negative. If I put in a value, let's say 00:41:10.760 --> 00:41:16.220 something like 4, then I get 16 - 12 - 2. Well that's positive, 00:41:16.220 --> 00:41:21.680 isn't it? 'cause it's an overall value of two. If I put in a 00:41:21.680 --> 00:41:26.360 value of, say something like let's say minus two, then I get 00:41:26.360 --> 00:41:32.598 4. Minus three times Y minus two is 6, so I get 10 takeaway 28. 00:41:32.598 --> 00:41:37.736 It's positive, so as I've gone from one value of X, I've got a 00:41:37.736 --> 00:41:42.140 positive value, then a negative and a positive value of X. So 00:41:42.140 --> 00:41:46.911 clearly it's going across the X axis at some point. So what I 00:41:46.911 --> 00:41:53.150 have to do is choose a value of X or a range of values of X that 00:41:53.150 --> 00:41:56.820 will enable me to graph that different behavior. First of 00:41:56.820 --> 00:41:58.288 all, positive values, negative 00:41:58.288 --> 00:42:00.700 values. Positive values again. 00:42:01.750 --> 00:42:07.980 So the range of values that I'm going to take, I'm going to take 00:42:07.980 --> 00:42:10.205 them going from minus 2. 00:42:11.520 --> 00:42:13.590 In unit steps. 00:42:14.100 --> 00:42:20.225 I'm going to go up to five. 00:42:21.330 --> 00:42:26.890 I'm going to calculate each term in turn. 00:42:27.520 --> 00:42:28.600 So let's just. 00:42:29.170 --> 00:42:32.334 Rule those off out of the way. 00:42:33.040 --> 00:42:39.555 X squared that will be 00:42:39.555 --> 00:42:46.070 4101, four, 916 and 25. 00:42:47.580 --> 00:42:51.408 The next term is minus 3X. 00:42:52.260 --> 00:42:56.866 So I have minus three times by the value of X, so minus three 00:42:56.866 --> 00:42:58.840 times by minus two is 6. 00:42:59.720 --> 00:43:03.696 Minus three times by minus one is 3. 00:43:04.780 --> 00:43:12.124 Minus three times by zero is 0 - 3 times by by 00:43:12.124 --> 00:43:19.468 one is minus three, and so on, minus 6 - 9 - 00:43:19.468 --> 00:43:21.304 12 - 15. 00:43:21.830 --> 00:43:27.940 Now we come to the last term minus two and that stays the 00:43:27.940 --> 00:43:33.110 same because it's the constant term, no matter what the value 00:43:33.110 --> 00:43:40.160 of X is, so we can just write in a line of minus Two's across 00:43:40.160 --> 00:43:46.270 our table. Now we can workout X squared minus three X minus two, 00:43:46.270 --> 00:43:51.910 because it's going to be the sum of each of these columns. 00:43:52.480 --> 00:43:56.812 So we have six and four is 10 takeaway, two is 8. 00:43:57.900 --> 00:44:05.370 One and three is 4 takeaway, two is 2 zero and zero and minus two 00:44:05.370 --> 00:44:11.844 is minus 2 - 3 and minus two is minus five and one. 00:44:11.990 --> 00:44:18.698 Minus 4 - 6 and minus two add 4. That's minus eight and 00:44:18.698 --> 00:44:21.278 four gives us minus 4. 00:44:22.850 --> 00:44:28.778 9 and minus nine is zero and minus two gives us minus 2. 00:44:29.540 --> 00:44:35.546 Minus 12 and minus two is minus 14 and 16 gives us +2. 00:44:37.330 --> 00:44:43.975 Minus 15 and minus two is minus 17. Add on the 25 and that gives 00:44:43.975 --> 00:44:49.291 us 8. Notice what we've got is symmetric across the table. Now 00:44:49.291 --> 00:44:51.949 that just happened to be because 00:44:51.949 --> 00:44:57.550 I picked. Values that were going to give me a complete symmetry, 00:44:57.550 --> 00:45:02.093 but you always find a degree of symmetry in the table. 00:45:02.760 --> 00:45:07.455 Now what I would need to do now is make a graph. I don't have 00:45:07.455 --> 00:45:10.898 any graph paper and so effectively what I'm going to do 00:45:10.898 --> 00:45:14.967 is. I'm going to draw a quick sketch because we know what the 00:45:14.967 --> 00:45:19.349 answer to this equation now, but I just want to show you how you 00:45:19.349 --> 00:45:21.227 would go about doing this so 00:45:21.227 --> 00:45:23.620 it's. Do that. 00:45:24.820 --> 00:45:27.220 We got out X axis. 00:45:28.430 --> 00:45:31.592 Our Y access you've got values 00:45:31.592 --> 00:45:34.870 minus 2. Minus one. 00:45:35.460 --> 00:45:43.496 12 3. Four and five, and I mark these 00:45:43.496 --> 00:45:45.688 often even numbers 2. 00:45:46.380 --> 00:45:54.090 4. 6. 8 and minus two 00:45:54.090 --> 00:45:56.850 and minus 4. 00:45:57.970 --> 00:46:03.410 So. Let's just flick back for a couple of values here. 00:46:03.990 --> 00:46:11.704 Minus two is 8 and minus one is 2, so I've got a point 00:46:11.704 --> 00:46:14.459 roughly there and a point 00:46:14.459 --> 00:46:21.976 roughly there. Then when it was zero, I had one here 00:46:21.976 --> 00:46:28.702 minus 2. And we go. And then at one and two they were 00:46:28.702 --> 00:46:32.680 both minus four, so they were there and there. 00:46:32.680 --> 00:46:38.608 Then at three and four, they were minus two and two, so they 00:46:38.608 --> 00:46:44.992 were there and there. And then at five it was back up here at 00:46:44.992 --> 00:46:50.464 8 again so I can get a nice smooth curve through these 00:46:50.464 --> 00:46:57.174 points. And then back up today. And where are my answers while 00:46:57.174 --> 00:47:03.596 there here, that one and that one? Those are the values of X 00:47:03.596 --> 00:47:09.524 for which is X squared minus three X minus 2 equals 0. 00:47:10.610 --> 00:47:13.805 Supposing, however, haven't been given that equation, but I've 00:47:13.805 --> 00:47:18.420 been given X squared minus three X minus 2 equals 6. Now, this 00:47:18.420 --> 00:47:23.035 shows the advantage of graphs I took a long time to work this 00:47:23.035 --> 00:47:28.360 out, and to draw this. And How do I draw on it neatly? Well, it 00:47:28.360 --> 00:47:32.620 would have taken even longer, but because I have drawn it, I 00:47:32.620 --> 00:47:37.235 can use all the data I've got, not only to solve the first 00:47:37.235 --> 00:47:40.785 equation, but to solve this equation as well, because all 00:47:40.785 --> 00:47:41.850 I'm looking for. 00:47:41.880 --> 00:47:46.677 Our values of X which make the function bit equal to six. So 00:47:46.677 --> 00:47:51.474 where is the function bit equal to 6? Where we go across here 00:47:51.474 --> 00:47:56.271 and we can see these are the points on the curve where it's 00:47:56.271 --> 00:48:01.806 equal to six and so if I come down, these are the values of X. 00:48:02.480 --> 00:48:07.030 Which correspond and I can just read off those values of X. So 00:48:07.030 --> 00:48:11.230 by drawing the graph I've not only been able to solve that 00:48:11.230 --> 00:48:15.080 one, but I've been able to solve this one as well. 00:48:15.320 --> 00:48:22.460 What about if I have X squared minus three X minus 5 equals 0? 00:48:23.580 --> 00:48:28.000 Can I use this graph again? Well, it's got the front bit the 00:48:28.000 --> 00:48:32.080 same. It's this bit that seems to be different, so if it's 00:48:32.080 --> 00:48:33.780 different, well, let's make it 00:48:33.780 --> 00:48:40.775 the same. So let's make it X squared minus three X minus 2. 00:48:42.030 --> 00:48:48.915 And then what do I need to keep it as minus five? Will I need 00:48:48.915 --> 00:48:54.423 minus three equals 0? So now this is my function bit the 00:48:54.423 --> 00:48:59.472 same, so I'm actually looking for when my function bit is 00:48:59.472 --> 00:49:03.603 equal to three. And again, that's within the possibilities. 00:49:03.603 --> 00:49:08.193 Here we have the value three, we can go across. 00:49:09.170 --> 00:49:15.092 There and then we can come down and read off the values of X, 00:49:15.092 --> 00:49:20.591 which gives us that. So for all drawing a graph took along time 00:49:20.591 --> 00:49:25.667 it actually enabled us to solve more than just the equation for 00:49:25.667 --> 00:49:27.359 which we drew it. 00:49:29.700 --> 00:49:31.940 So let's just recap. 00:49:32.670 --> 00:49:37.260 There's a basic quadratic equation X squared plus BX plus 00:49:37.260 --> 00:49:42.768 C equals 0. There are four ways of solving it, first by 00:49:42.768 --> 00:49:46.440 Factorizing and you should try that way first. 00:49:46.970 --> 00:49:50.822 Second, by completing the square, but that's not advised, 00:49:50.822 --> 00:49:55.958 so to speak. Third, if it won't factorize, use the formula and 00:49:55.958 --> 00:50:00.238 those two ways factorizing and using the formula. Other prime 00:50:00.238 --> 00:50:04.518 ways. The first methods of recourse that you should use. 00:50:04.518 --> 00:50:08.798 Finally, just drawing a graph. Time consuming but may be 00:50:08.798 --> 00:50:13.934 beneficial if you've got a lot of the same kind of equation 00:50:13.934 --> 00:50:16.074 that you need to solve.