[Complex variables] z^4 - 4z^2 +4 -2i = 0

So let u=z2, then...u2-4u+4=2i(u-2)2=2i+4u-2=+-√(2i+4)u=2+-√(2i+4)Now substitute back for u=z2z2=2+-√(2i+4)Now solve for z. Since this is a quadratic equation, there will be two solutions for z for each value of u, so in total there will be 4 solutions for z.
  • #1
kingwinner
1,270
0

Homework Statement


"This is an example from my textbook:
Solve the equation z4 - 4z2 + 4 - 2i = 0

Solution:
Rearranging, we get z4 - 4z2 + 4 = 2i
or (z2 - 2)2 = 2i = (1+i)2
This has solutions z2 - 2 = 1+i or -1-i.
Equivalently z2=3+i or z2=1-i

These may be solved to give the 4 solutions of the original equation.
cv1.JPG
"
==========================

I don't understand the following step:
(z2 - 2)2 = (1+i)2 => z2 - 2 = 1+i or -1-i

Why is this true? I remember for real numbers we have √(x2) = |x| (note that it is |x|, not x). Is this true for complex numbers? If so, then
(z2 - 2)2 = (1+i)2
=> z2 - 2 = +/- √[(1+i)2] = +/- |1+i| ?

Homework Equations


N/A

The Attempt at a Solution


Shown above.

I hope someone can explain this. Any help is appreciated!
 
Last edited:
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  • #2
Make this substitution: z^2 = t
and solve by the quadratic formula: t^2 - 4t + (4 - 2i) = 0
get the answers, go back to the substitution and get in total 4 roots (some roots may be the same).
 
  • #3
kingwinner said:
I don't understand the following step:
(z2 - 2)2 = (1+i)2 => z2 - 2 = 1+i or -1-i

Why is this true? I remember for real numbers we have √(x2) = |x| (note that it is |x|, not x). Is this true for complex numbers?
That is true for real numbers.

However, you may have forgotten your equation solving: if you know that x2=a2, then x could be either a or it could be -a. In the real numbers, if you solved this equation by taking the (real) square root of both sides, you get |x|=|a|, not x=|a|.

Anyways, in the complexes, x2 has two square roots, just as in the reals. Note that |x| usually isn't one of them. The principal square root (the analog of the real √ function) is the square root that either has positive imaginary part, or lies on the non-negative real axis. The other square root is its negative, as usual.

Incidentally, even in the complexes, if you know that x2=a2, then it is also true that |x|=|a|. However, you can't do much with that, because there are lots of possible values for x, but only 2 of them satisfy the original equation
 
  • #4
I see.

Then how can we actually justify or rigorously prove that
(z2 - 2)2 = (1+i)2 => z2 - 2 = 1+i or -1-i ??
 
  • #5
kingwinner said:
I see.

Then how can we actually justify or rigorously prove that
(z2 - 2)2 = (1+i)2 => z2 - 2 = 1+i or -1-i ??

Because we know (1+i)2 has two square roots, and its obvious that those are them. Similarly, the two square roots of (z2-2)2 are obvious.

Alternatively, bring them together and factor.
 
  • #6
But can we factorize/expand as usual when z and c are COMPLEX numbers??
(z-c)(z+c) = z^2 - c^2

If so, why?
 
  • #7
Why can you do it for real numbers? The same reason applies here.
 
  • #8
For real numbers, it's true because we can expand it; I think it's called distributive law.

Does this still hold for complex numbers z and c?
 
  • #9
kingwinner said:

Homework Statement


"This is an example from my textbook:
Solve the equation z4 - 4z2 + 4 - 2i = 0

Solution:
Rearranging, we get z4 - 4z2 + 4 = 2i
or (z2 - 2)2 = 2i = (1+i)2
This has solutions z2 - 2 = 1+i or -1-i.
Equivalently z2=3+i or z2=1-i

These may be solved to give the 4 solutions of the original equation.
cv1.JPG
"
==========================

I don't understand the following step:
(z2 - 2)2 = (1+i)2 => z2 - 2 = 1+i or -1-i

Why is this true? I remember for real numbers we have √(x2) = |x| (note that it is |x|, not x). Is this true for complex numbers? If so, then
(z2 - 2)2 = (1+i)2
=> z2 - 2 = +/- √[(1+i)2] = +/- |1+i| ?
What you need to remember is not [itex]\sqrt{x^2}= |x|[/itex] (even for complex x, |x| is positive real number) but rather that the equation [itex]x^2= a^2[/itex] has the two roots x= a and x= -a. And, yes, that is true for complex numbers!

(A little more about [itex]\sqrt{x^2}= |x|[/itex]. Working in the real numbers, we like functions to have one value for a given value of the argument- in fact, that is part of the definition of "function". That's why we define \(\displaystyle \sqrt{a}\) to be the positive number whose square is a: \(\displaystyle \sqrt{x^2}= |x|. But in the complex numbers we cannot do that- the complex numbers do not form an "ordered field" and, in particular, we cannot talk about "positive complex numbers". We have to accept "multi-valued functions" and, in fact, almost all functions, even those that are simple functions in the real numbers, become mult-valued functions over the complex numbers.)

Homework Equations


N/A

The Attempt at a Solution


Shown above.

I hope someone can explain this. Any help is appreciated!
\)
 
  • #10
kingwinner said:
For real numbers, it's true because we can expand it; I think it's called distributive law.

Does this still hold for complex numbers z and c?
The distributive law? Yes...

(And the distributive law is easy to prove if you weren't sure about it)
 
  • #11
The method used in the example seems to apply only in very special cases.
Is there a more general way to solve z4 - 4z2 + 4 - 2i = 0 ? Can we use the quadratic formula here? If so, can someone please outline the steps of solving this problem using the quadratic formula? (because I don't quite understand how to use it here and how we are supposed to get FOUR solutions this way)

Thanks!
 
  • #12
The example solution is pretty much another form of the quadratic formula. They instead just completed the square. But if you want to try it out, just let u=z2 and solve for u, then substitute back
 

Related to [Complex variables] z^4 - 4z^2 +4 -2i = 0

1. What is a complex variable?

A complex variable is a mathematical quantity that has both a real and an imaginary component. It is written in the form z = x + iy, where x and y are real numbers and i is the imaginary unit equal to the square root of -1.

2. How do you solve a complex variable equation?

To solve a complex variable equation, we can use algebraic methods to manipulate the equation into a form where we can isolate the variable z on one side. We can then use the quadratic formula or other methods to solve for the values of z.

3. What is the significance of the equation z^4 - 4z^2 +4 -2i = 0?

This equation is known as a quartic equation, which is a type of polynomial equation with a degree of 4. It has four complex solutions, which can be found by solving for the roots of the equation.

4. How can complex variables be applied in science?

Complex variables are used in various fields of science, including physics, engineering, and mathematics. They can be used to model and analyze systems with both real and imaginary components, such as electrical circuits, fluid dynamics, and quantum mechanics.

5. Is there a real-life application of this specific complex variable equation?

Yes, this equation has real-life applications in electrical engineering and signal processing. It can be used to model the behavior of a circuit with a complex impedance, or to analyze a signal with both real and imaginary components. It can also be applied in other fields, such as optics and wave propagation.

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