A Rational Equation With Complex Numbers | P643

[00:00:00]Hello everyone, welcome to a plus bi.

[00:00:03]This channel is all about complex numbers. And in this video, we're going to be solving a very nice equation with complex numbers. This is actually a rational equation, but we're going to turn it into a polomial and find all the solutions. All right, so let's go ahead and get started. If you're new to complex numbers, go ahead and check out my lecture videos. Uh you can also check out my other channel, which is Cybermath. I publish Mondays and Fridays, two videos per week. It's all about trigonometry, complex, well, sometimes complex numbers, trigonometry, algebra, and number theory mainly.

[00:00:39]Great. So, let's go ahead and see how we can solve an equation like this. Now, first of all, notice that there's nothing factoable in the numerator or denominator that will factor easily. So, we're going to go ahead and cross multiply. In other words, we're going to multiply both sides by Z ^2 - 2 Z + 2.

[00:00:58]Okay? And that gives us the following. Z ^2 + 2 Z + 2. That cancels out when we multiply. And when you multiply Z by that, we get Z cub minus 2 Z ^ 2 + 2 Z.

[00:01:12]2 Z or not 2 Z. I always make that cheap joke. Now, we're going to go ahead and get rid of the 2Z because they are the same. We don't need to worry about it.

[00:01:21]Now we're going to put everything on the same side and that'll give us z cubed.

[00:01:25]I'm going to bring the z ^2 over and then switch sides. Minus 3 z ^2 and then of course that's going to be since I'm collecting everything on the right hand side it'll be a minus2 = z. Great. Now we did get a cubic equation didn't we?

[00:01:42]And can we solve cubic equations? Is there a cubic formula? Definitely there is a cubic formula and I use that a lot even though I don't write it as a formula. I'll show you how it's done so that you can remember the process because if you just if I just give you a formula you're going to memorize and you'll probably forget it sooner or later. Now to solve this problem first of all we're going to make it depressed which means we do want this not to have anything squared like Z squared. Of course we're going to change the variable so we don't want any quadratic terms. Make sense? And to be able to do that, we're going to set Z equal to something like W + 1. And let me tell you where that one comes from. How do you know that? Well, you look at the coefficient of Z ^2, which is -3. And you negate that number and divide by the degree, which is three. In other words, you take this number and divide by -3, which gives you one. And then you just add that to another variable, and that's going to be your new thing. So now we're going to replace Z with W + one everywhere. That's going to give us a really nice cubic equation. You'll be amazed how nice that is. Now let's go ahead and replace Z with W + 1 cubed - 3 * W + 1^ 2 - 2. If you want, you can also set it equal to 2. No big deal.

[00:02:59]We're going to have to collect everything at the end anyways. Now we're going to have to expand this. So, it's going to be w cubed + 3 w ^2. Since this is well known, I can just do it the normal way. And then minus 3 * w ^2 + 2 w + 1. You got to memorize these formulas. And then it's going to equal 2. No big deal. We're going to have to add all the constants at the end. Now, this gives us w cub + 3 w ^2 + 3 w + 1 and then distribute the -3. And you will notice that something cancels out. Okay, great. Now take a look. W cub 3 W ^2 W ^2 cancel out. And that was our goal.

[00:03:42]That's why this is the right way to do it. And if you don't know how to do it, you can say, hey, I want to replace Z with W + H and then to make W ^2 0, what's the H value? And you're going to end up with H equals 1. Make sense?

[00:03:56]That's a way to find out how this is done if you don't memorize it. But that's easy to memorize hopefully. Okay, great. Now, what are we going to do?

[00:04:05]Let's kind of put it together. W cubed, we have 3 - 6, that's at - 3 W. We have a 1 - 3, which is a -2. And I'll put that on the right hand side by adding two to both sides. This is going to turn into a four. Great. Well, this is still a cubic equation. So, should I just solve it with the cubic formula? Because we haven't used it yet, right? We only made the equation depressed. We depressed this equation. Now it's in depression. Okay. Now we can use the formula. Uh but it's always worth checking rational root theorem which means if you put it in the standard form any number that divides four could be a rational solution. And those numbers are usually 1 2 4 and they're negatives.

[00:04:51]Right? So you can kind of check it out.

[00:04:53]One is not going to work. Two is not going to work. We can check it out. 8 - 6 - 4. Nope. That's not zero. If you check four, four is going to make W cube bigger. So, it's not going to work either. Negative one is not going to work. Well, it looks like it's going to work. For example, if we had something like this, maybe it would work. I don't know. Or something like this. I think one would work in this case, right? But we don't have any of those. So, negative 1 doesn't work either. How about -2? Uh, because I have to test all of them, right?

[00:05:25]-2 is going to give me8 + 6 - 4. Uhoh, that doesn't work either. And obvious is not going to work. So there are no rational solutions. We can safely say that there are no rational solutions. So what are we supposed to do? Use the cubic formula. So I'm going to go ahead and go back to this format. Okay?

[00:05:43]Because what I'm about to write is hopefully going to help you remember how this is done. We're going to use a + b cubed minus 3 a * a + b. This is the formula that I nor normally use for sum of two cubes. But guess what? This gives you the cubic formula. How? If you assume that okay a plus b is w then you do get another cubic equation. One of whose solutions is a plus b, right? So what does that mean? If I can compare this equation to mine and find the values of a and b, then I can find w, right? Or what I'm looking for. To be able to do that, we're basically going to compare these two equations, which means a is equal to 1 and a cub + b cubed is equal to 4. Awesome. What does that give us? Are there any rational solutions? Probably not. So, let's go ahead and u solve this system. I'm going to cube both sides here and then replace b cubed with 4 - a cubed and substitute here. Okay, so it's going to give me a cub * 4 - a cub = 1. And you got to remember this is not going to be a hexic or a cubic equation. This is going to turn into a quadratic. Why? If you set a cub= to c, and there's a good reason because we need b at the end. Now you're going to get c * 4 - c = 1, which gives you c ^ 2 - 4 c + 1 = 0. Easy, right?

[00:07:08]Now you turn the qu cubic into a quadratic which is nice because we know how to solve quadratics. The formula is fairly simple. Now you can use either completing the square subtract one and add four or add three to both sides or you can just use the formula. Let's go ahead and use the completing the square method. Add three to both sides. You get four here and three here. Now notice that the left hand side is a perfect square because we added the right term.

[00:07:33]And now we can square root both sides. C minus 2 would be plus -<unk> 3 and from here C would be 2 + -<unk>3.

[00:07:42]But wait a minute, what is C? C is A or B, right? No, not really actually.

[00:07:49]C is A cubed. So this is equal to A cubed or B cub b cubed because there were two roots remember. So if you set it equal to A cubed, then A would be the cube root of this. In other words, a can be something like cube root of this and then b can be the other one or vice versa. But at the end, it doesn't matter. You know why? Because we're looking for a plus b. If you remember, we set w equal to a plus b. So if you want to solve for w, it'll be a plus b, which is the cube root of 2 + roo<unk>3 plus the cube root of 2 minus roo<unk>3.

[00:08:24]How nice, right? A very radical expression. Well, that's what it is.

[00:08:29]Now, this is irrational and real obviously, right? It's a real number.

[00:08:34]Well, are there any other real solutions? We'll find out. So, but that's not the end because we're not looking for W. If you remember, W comes from where? Let's go back. Go back. Go back. W + 1 is equal to Z. In other words, W is equal to Z minus one. In other words, well, Z is W + 1. I guess I'm confusing myself here. So Z is going to be this number plus one, right? And we were looking for Z, weren't we? So the answer at least one of the roots will be this one. Now how do you find the other solutions? Well, well, you can make this a polomial like a linear one Z minus let's just call this Z sub0. You can just turn this into Z sub0 and take your polomial and divide by this.

[00:09:23]That'll give you a quadratic and then you can solve it. That's way too complicated. Don't do it. Because here's the catch. If you are solving a cubic equation and you solved it, let's say Z sub0 is one of the solutions, right? So here's what happens. You can multiply this by cube roots of unity. Okay? Which are usually W and W ^2. But this is not the same W I'm talking about. So sorry about the confusion. I should probably use a different variable like y maybe.

[00:09:55]But the idea is multiply this by cube roots of unity.

[00:10:00]Cube roots of unity except for one. Of course, you can also multiply by one, but it's going to give you the same solution. By the way, Z sub0 is this one. And then you'll get the other solutions. What's a cubit of unity?

[00:10:12]Let's talk about that. E to the power 2 pi ni, as you know, uh is how one is written, right? In the complex word. If you divide uh or raise both sides to the power 1/3, you get cube roots of unity.

[00:10:26]For example, if n is equal to 1, then from here you get e to the power 2<unk>i i / 3, which is the same as cosine 2<unk>i / 3 + i sin 2 pi. Let me show you one of them and hopefully you can find the other one which will give you for n= 2. Now if you take this solution the cube<unk> of 2 + <unk>3 plus cube<unk> of 2 minus <unk>3 + one remember this was the first z solution we found and multiply by cosine of 2<unk> over 3. What is cosine of 2 p<unk> 3 by the way? Let's find it on the unit circle. 2 p<unk> over 3 is basically 120°. So like 90 + 30 is it cosine is going to be negative in the second quadrant but it's going to be like this uh like cosine 60 which is 12.

[00:11:14]So we're going to get something like 12 plus <unk>3 over 2 i. And of course the other one is going to be the 240 which is going to be here and it'll have both the s and cosine as negatives. And you can pretty much do the same thing. But when you multiply them together, you're gonna get a beautiful, beautiful complex solution. Right? Don't you love that? Go ahead and do it and simplify and share with us in the comment section because we are nearing the end. Now I'm going to show you what wolf from alpha provides as the solutions because I also asked Wolf from alpha, hey, can you solve this equation? Of course, it can because it's a basic cubic and ta da the real solution agrees with us. But the cubic solution or the complex solutions, they don't look very good, do they? You can go ahead and verify this with the given information. And this brings us to the end of this video. Thank you for watching. I hope you enjoyed it. Please let me know. Don't forget to comment, like, and subscribe. I'll see you next time with another video. Until then, be safe. Take care. Don't forget to check out Cybermath and A+BI. And bye-bye.