Solving A Complex Quartic

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Solving A Complex Quartic | Problem 284

Рет қаралды 804

Ай бұрын

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Пікірлер: 17

@MAZHOR1113 Ай бұрын

4th metod is of course the damn quartic formula

@scottleung9587 Ай бұрын

Yeah, good luck using that!

@Qermaq Ай бұрын

3:37 I see the distinction this way. By solving for w or for z^2, we are basically solving for the square of our value. So that's like when you square something along the way and then take the square root, you have to consider both square roots in that case because you maybe introduced an extraneous solution. But in this case it's even easier. It's a quartic. There are four answers so we have to use both square roots of the values for w or z^2.

@aplusbi Ай бұрын

Good thinking!

@scottleung9587 Ай бұрын

I used the first method.

@Qermaq Ай бұрын

Hmm, wondering if there's more to it than in the thumbnail. As is, it's a quadratic in z^2, pretty routine really. z^2 = 4 or -1, so the four solutions should be 2, -2, i and -i.

@phill3986 Ай бұрын

😊😊😊👍👍👍

@mcwulf25 Ай бұрын

Method 4 Observe z^2 = 4 is a solution. Factor this out (z^2 - 4)(z^2 + 1) = 0 And the solutions drop out.

@SweetSorrow777 Ай бұрын

How's using euler's formula and writing in polar form as a 4th method?😊

@aplusbi Ай бұрын

Can you give some steps?

@ChrisStoneinator Ай бұрын

Disgusting, that’s what it is

@seanfraser3125 Ай бұрын

z^4 - 3z^2 - 4 = 0 (z^2 - 4)(z^2 + 1) = 0 (z-2)(z+2)(z-i)(z+i) = 0 Our four solutions are z=2, z=-2, z=i, and z=-i

@StuartSimon Ай бұрын

I was originally going to use the quadratic formula to solve for z^2, but then I realized that I could easily factor twice.

@Qermaq Ай бұрын

6:00 sorry I'm chatty tonight. Doesn't 0 have two square roots, itself and itself? I mean, we can have a quadratic like x^2 + 2x + 1 = 0 and the answer is often given as 1 (multiplicity 2) so I think of sqrt(0) the same way, and indeed nroot(0). Yes, the principal root is 0, but the secondary roots are also 0. It's a kinda pointless distinction except for the idea that a n-tic equation must have n values.

@aplusbi Ай бұрын

Good question. 0 is an interesting number. In the complex world, it's the zero vector that has no direction. What's its argument? I think it's 0 but tan(arg(0)) is undefined. very problematic...

@aplusbi Ай бұрын

Hey! nothing wrong with that. Actually, it's highly encouraged, wink, wink 😄