WEBVTT 00:00:03.050 --> 00:00:07.756 In this unit, we'll see how we can use the imaginary number I 00:00:07.756 --> 00:00:09.566 to solve any quadratic equation. 00:00:10.100 --> 00:00:15.872 Let's have a look at an example. Suppose we want to solve the 00:00:15.872 --> 00:00:17.648 quadratic equation X squared 00:00:17.648 --> 00:00:20.834 minus 2X. Plus 10 00:00:20.834 --> 00:00:25.990 equals 0. Now we're going to use the formula for solving a 00:00:25.990 --> 00:00:29.675 quadratic equation. This is the formula over here and first of 00:00:29.675 --> 00:00:33.695 all we need to identify the values of AB&C to substitute in 00:00:33.695 --> 00:00:38.050 the formula. Now the value the value of a is the coefficient of 00:00:38.050 --> 00:00:40.395 X squared, which in this case is 00:00:40.395 --> 00:00:45.808 one. The value of B is the coefficient of X, which is minus 00:00:45.808 --> 00:00:50.006 2. And the value of C is the constant term, which is 10. 00:00:50.010 --> 00:00:55.890 Can we substitute these values into this formula? So here we go 00:00:55.890 --> 00:01:01.350 X equals. Minus B, which is minus minus two, which is +2. 00:01:01.980 --> 00:01:07.916 Plus or minus the square root of B squared, B squared is minus 2 00:01:07.916 --> 00:01:09.612 squared, which is +4. 00:01:10.180 --> 00:01:15.208 Minus four times a, which was one and see which was 10. 00:01:15.930 --> 00:01:20.649 All divided by 2A and 2A is 2 one or two. 00:01:21.990 --> 00:01:26.358 Let's tidy this up. We've got 2 plus or minus. Now let's look 00:01:26.358 --> 00:01:30.726 under the square root sign. We've got 4. Subtract 4 * 1 * 00:01:30.726 --> 00:01:36.940 10. 4 * 1 * 10 is 44. Subtract 40 is minus 36, so you'll see 00:01:36.940 --> 00:01:40.658 we've got a square root of a negative number. Here. The 00:01:40.658 --> 00:01:44.376 square root of minus 36, and it's all divided by two. 00:01:45.820 --> 00:01:50.123 Let me remind you how you deal with the square root of a 00:01:50.123 --> 00:01:54.426 negative number. The square root of minus 36. We can write as the 00:01:54.426 --> 00:01:56.412 square root of 36 times minus 00:01:56.412 --> 00:01:58.910 one. Which is 6. 00:01:59.420 --> 00:02:02.240 Times I or six I. 00:02:03.470 --> 00:02:06.781 So the square root of this negative number, the square root 00:02:06.781 --> 00:02:08.587 of minus 36, simplifies to just 00:02:08.587 --> 00:02:14.973 six I. And finally, if we just want to tidy this up a bit more, 00:02:14.973 --> 00:02:19.785 we can notice that there's a factor of two in the numerator 00:02:19.785 --> 00:02:23.795 and the denominator, which can be cancelled, which will leave 00:02:23.795 --> 00:02:29.008 one plus or minus three. I, so here we have two solutions of 00:02:29.008 --> 00:02:33.820 the quadratic equation. One of the Solutions is is the number 1 00:02:33.820 --> 00:02:38.632 + 3 I and another is the number 1 - 3 I. 00:02:39.390 --> 00:02:42.910 Let's have a look at another example. In this example, we're 00:02:42.910 --> 00:02:46.430 going to study the quadratic equation, two X squared plus X 00:02:46.430 --> 00:02:47.710 Plus One is 0. 00:02:48.680 --> 00:02:54.568 Again, in order to 00:02:54.568 --> 00:02:57.512 use the 00:02:57.512 --> 00:03:03.577 formula. Which is here we need to identify the values of AB&C. 00:03:04.570 --> 00:03:09.458 The value of a, which is the coefficient of X squared, is 2. 00:03:10.340 --> 00:03:15.954 The value of B is one and the value of C is also one. 00:03:17.080 --> 00:03:19.960 And we substitute these values into the formula. 00:03:20.610 --> 00:03:27.750 So we'll get X equals minus B, which is minus one plus or minus 00:03:27.750 --> 00:03:30.810 the square root of be squared. 00:03:31.560 --> 00:03:33.600 Which is 1 squared. 00:03:34.130 --> 00:03:40.396 Minus four times a which was two and see which was one all 00:03:40.396 --> 00:03:44.636 divided by. To a which is 2 twos of four. 00:03:45.690 --> 00:03:50.142 Let's tidy it. What we've got minus one plus or minus. Now 00:03:50.142 --> 00:03:54.594 let's look at the square root. We've got 1 squared, which is 00:03:54.594 --> 00:03:58.675 one subtract 428, so it's one subtract 8, which is minus 00:03:58.675 --> 00:04:02.756 Seven. You'll see again that we've ended up with a square 00:04:02.756 --> 00:04:04.611 root of a negative number. 00:04:06.110 --> 00:04:10.244 Now the square root of minus Seven we handle in the same way 00:04:10.244 --> 00:04:14.378 as before. We write it as the square root of 7 times minus 00:04:14.378 --> 00:04:18.194 one. The square root of Seven relievers. The square root of 7 00:04:18.194 --> 00:04:22.010 and the square root of minus one. We now right as I. 00:04:23.040 --> 00:04:27.840 So this solution we have here now simplifies to minus one plus 00:04:27.840 --> 00:04:32.640 or minus the square root of minus Seven. We write as the 00:04:32.640 --> 00:04:34.240 square root of 7. 00:04:34.430 --> 00:04:38.928 Times, I and the whole things divided by 4 and we can leave 00:04:38.928 --> 00:04:43.772 our answer like that. But if we want to we can write it as 00:04:43.772 --> 00:04:47.578 separate terms. We can write it is minus 1 / 4. 00:04:48.110 --> 00:04:49.448 Plus or minus. 00:04:50.210 --> 00:04:52.750 The square root of 7. 00:04:52.750 --> 00:04:56.880 Divided by 4 multiplied by I. So either of those 00:04:56.880 --> 00:04:58.119 forms are equivalent. 00:04:59.890 --> 00:05:04.048 We've now seen how we can write down the solution of 00:05:04.048 --> 00:05:05.182 any quadratic equation. 00:05:06.430 --> 00:05:11.470 A number such as this one which has got a part which is purely 00:05:11.470 --> 00:05:15.790 real in this case, minus 1/4 and a part which is imaginary. 00:05:15.790 --> 00:05:20.110 That's the part that this number here. Route 7 over 4 multiplied 00:05:20.110 --> 00:05:24.790 by this imaginary number I a number such as this is called a 00:05:24.790 --> 00:05:28.750 complex number, and in the next unit will define properly what 00:05:28.750 --> 00:05:30.910 we mean by a complex number.