Finding Roots of complex numbers (solve non-linear multi-variable system of equation)

In summary: This would make the problem much simpler.In summary, the problem involves solving for the roots of two equations using algebraic methods. The first equation is solved by breaking down the second equation into two parts and using substitution. For the second equation, the professor gave a hint to make both sides equal to 12 and subtract them to form a new equation. However, it seems that the student's algebra skills may need improvement and they are unsure how to proceed. They are avoiding using complex exponential form, which would simplify the problem.
  • #1
Koolaidbrah
3
0

Homework Statement


1. z^6=(64,0)
2. z^4=(3,4)


Homework Equations


These are expanded out into Real and Imaginary components (treat them seperate):
1. REAL (EQ 1) - x^6-15x^4y^2+15x^2y^4-y^6=64
IMAG (EQ 2) - 6x^5y-20x^3y^3+6xy^5=0
From here, you basically solve these for all six roots.
2. REAL (EQ 1) - x^4-6x^2y^2+y^4=3
IMAG (EQ 2) - 4x^3y-4xy^3=4


The Attempt at a Solution


These must be done algebraically, not using Euler angle components (answer would be trivial)

For #1, I have EQ 2 broken down into
3z^2+3v^2=10zv where z=x^2 and v=y^2.
I know the solution is z= 3z and z=3/z, which I can then plug into EQ 1, and all my answers will be given. My algebra is just lacking to get those 2 answers for z.

#2 is a lot more difficult and there is no 'zero equation'. Professor gave us hint to make both sides =12 at the end, subtract them to get an equation with 0; this gives me 3 equations

EQ 2 - xy(x^2-y^2)=1
EQ 1 - x^4+y^4=3(1+2x^2y^2)
EQ 3 (new formed, what I will likely end up using to solve x in terms of y) - x^4-3x^3y-6x^2y^2+3xy^3+y^4=0

My algerbra must be lacking. Not looking for a given answer, but any hints that will help me solve these non-linear multi-variable system of equations
 
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  • #2


Is there some reason you're avoiding using the polar (complex exponential) form of complex numbers?
 

1. What are complex numbers and why do we need to find their roots?

Complex numbers are numbers that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. They are necessary for solving certain mathematical equations that cannot be solved with real numbers alone. Finding the roots of complex numbers allows us to solve non-linear multi-variable systems of equations, which have many real-world applications in fields such as physics, engineering, and computer science.

2. How do you find the roots of complex numbers?

To find the roots of complex numbers, we use a method called the quadratic formula. This formula allows us to solve equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. By substituting the values of a, b, and c from our complex number equation into the quadratic formula, we can find the roots of the equation.

3. What is the difference between real and complex roots?

Real roots are solutions to an equation that are real numbers, meaning they do not involve the imaginary unit i. Complex roots, on the other hand, involve the imaginary unit and are expressed in the form a + bi. This means that the solutions to an equation with complex roots will have both a real and imaginary component.

4. Can complex numbers have multiple roots?

Yes, complex numbers can have multiple roots. This means that there can be multiple solutions to an equation with complex roots. For example, the equation x^2 + 1 = 0 has two complex roots: i and -i. This is because when we substitute these values into the equation, they both make the equation true.

5. How are complex roots represented on a graph?

Complex roots are represented on a graph using the complex plane, which is a two-dimensional coordinate system where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part. The complex roots of an equation are represented as points on this plane, with their real and imaginary components as their coordinates. For example, the complex root i would be represented as the point (0,1) on the complex plane.

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