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

[kingwinner]

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.

"
==========================

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!

 

[gomunkul51]

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).

 

[Hurkyl]

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

 

[kingwinner]

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 ??

 

[Hurkyl]

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.

 

[kingwinner]

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?

 

[Hurkyl]

Why can you do it for real numbers? The same reason applies here.

 

[kingwinner]

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?

 

[HallsofIvy]

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.

"
==========================

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 \sqrt{x^2}= |x| (even for complex x, |x| is positive real number) but rather that the equation x^2= a^2 has the two roots x= a and x= -a. And, yes, that is true for complex numbers!

(A little more about \sqrt{x^2}= |x|. 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 [math]\sqrt{a}[/math] to be the positive number whose square is a: [math]\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!

 

[Hurkyl]

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)

 

[kingwinner]

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!

 

[Mentallic]

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