Question:How can I solve the complex equation z^4-2z^2+4=0?

In summary, the conversation is about solving a complex number equation: z^4-2z^2+4=0. The solution is +-1/2(\sqrt{6}+-\sqrt{2i}) in four combinations of signs. The conversation then discusses different methods of solving the equation, including factoring and bisecting. It is eventually determined that the solution involves taking the square root of z^2 and using the quadratic formula, leading to the solutions z=+-\sqrt{1+i\sqrt{3}} and z=+-\sqrt{1-i\sqrt{3}}. The conversation concludes with a question about where the given solution came from.
  • #1
JasonRox
Homework Helper
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Complex Number, again... :(

This I'll give you the entire question and answer.

10. Solve:

[tex]z^4-2z^2+4=0[/tex]

That's all I got.

Answer:

[tex]+-1/2(\sqrt{6}+-\sqrt{2i}) (four combinations of signs).[/tex]

That is all.

I tried factoring, but I can't come up with anything. I also tried bisecting, but that is useless.

Knowing that the solution has complex numbers, I have no clue where they got this from.
 
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  • #2
solve for [tex] z^{2} [/tex] then take the square root of those...
 
  • #3
Ok then.

[tex]z^2=(\frac{z^2}{\sqrt{2}}-2i)(\frac{z^2}{\sqrt{2}}+2i)[/tex]

The square root of these?
 
  • #4
my results are
[tex]z=+-\sqrt{1+i\sqrt{3}}[/tex]
[tex]z=+-\sqrt{1-i\sqrt{3}}[/tex]

I don't know whether they are correct or not..
 
  • #5
let x= z2 then the equation z4- 2z2+ 4= 0 becomes x2- 2x+ 4= 0. That is the same as x2- 2x= -4 or
x2- 2x+ 1= (x- 1)2= -3. From that x= [itex]1+- \sqrt{3}i[/itex].
Since x= z2, [itex]z^2= 1+- i\sqrt{3}[/itex] so [itex]x= +- \sqrt{1+- i\sqrt{3}}.
 
  • #6
I can see that, but where would their solution come from?

Using the quadratic formula yield the same result as yours, too.
 

1. What are complex numbers?

Complex numbers are numbers that have both a real and an imaginary part. They are represented in the form a + bi, where a is the real part and bi is the imaginary part (b is a real number and i is the imaginary unit, equal to the square root of -1).

2. How are complex numbers different from real numbers?

Complex numbers have both a real and imaginary part, while real numbers only have a real part. This allows complex numbers to represent quantities that cannot be expressed using real numbers, such as the square root of a negative number.

3. What is the purpose of using complex numbers?

Complex numbers are used in various fields of mathematics and science, such as engineering, physics, and signal processing. They are particularly useful in representing and solving problems involving oscillations, electrical circuits, and quantum mechanics.

4. How do you perform operations with complex numbers?

To add or subtract complex numbers, you simply add or subtract the real and imaginary parts separately. To multiply complex numbers, you use the FOIL method (first, outer, inner, last) and simplify using the fact that i squared is equal to -1. To divide complex numbers, you multiply the numerator and denominator by the complex conjugate of the denominator.

5. Can complex numbers be graphed on a coordinate plane?

Yes, complex numbers can be graphed on a coordinate plane known as the complex plane. The horizontal axis represents the real part and the vertical axis represents the imaginary part. This allows for visualizing and geometrically interpreting complex numbers and their operations.

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