Solving the Complex Equation Z4 - 16i Z2 – 3 sqrt3 i 0
" "When dealing with complex equations, it's important to understand the underlying principles and techniques. In this guide, we will walk you through the process of solving the complex equation Z4 - 16i Z2 – 3 sqrt3 i 0. This equation has already been partially factorized, making the process more straightforward. We will break down the problem into simpler steps, explaining each one to ensure you understand the solution.
" "Factorized Form
" "Let's start with the factorized form of the given equation:
" "Z4 - 16i Z2 – 3 sqrt3 i 0" "
This can be factored into:
" "" "Z4 16i" "Z2 3 - sqrt{3}i" "" "Solving Z4 16i
" "The equation Z4 16i can be solved by finding the fourth roots of 16i. To do this, we first express 16i in polar form. Let's break it down:
" "16i 16e(pi/2)i (since 16 has no real component and its argument is pi/2).
" "The fourth roots of 16i are given by:
" "z 2e[(pi/2 2kpi)/4]i for k 0, 1, 2, 3" "
Substituting k 0, 1, 2, 3:
" "" "k 0: z 2e(pi/8)i" "k 1: z 2e(5pi/8)i" "k 2: z 2e(9pi/8)i" "k 3: z 2e(13pi/8)i" "" "These roots are equally spaced around a circle, as you would expect from the properties of roots of unity. The principal root (k 0) has an argument of pi/8.
" "Solving Z2 3 - sqrt{3}i
" "Next, we solve the equation Z2 3 - sqrt{3}i. To find the square roots of 3 - sqrt{3}i, we again express it in polar form:
" "3 - sqrt{3}i 2sqrt{3}e(-pi/6)i (since the magnitude is sqrt(3^2 (sqrt{3})^2) 2sqrt{3} and the argument is -pi/6).
" "The square roots are given by:
" "z sqrt{2sqrt{3}}e[-pi/12 kpi/2]i for k 0, 1" "
Substituting k 0, 1:
" "" "k 0: z sqrt{2sqrt{3}}e(-pi/12)i" "k 1: z sqrt{2sqrt{3}}e(5pi/12)i" "" "These roots are also equally spaced around a circle. The principal root (k 0) has an argument of -pi/12.
" "Conclusion
" "In summary, the solutions to the equation Z4 - 16i Z2 – 3 sqrt3 i 0 are given by:
" "" "The fourth roots of 16i: z 2e(pi/8)i, z 2e(5pi/8)i, z 2e(9pi/8)i, z 2e(13pi/8)i" "The square roots of 3 - sqrt{3}i: z sqrt{2sqrt{3}}e(-pi/12)i, z sqrt{2sqrt{3}}e(5pi/12)i" "" "Each root is equally spaced around a circle, and the principal roots have arguments of pi/8 and -pi/12, respectively.
" "Frequently Asked Questions
" "What is the significance of the roots of unity?
" "The roots of unity are the solutions to the equation zn 1. They have a cyclical and symmetrical distribution on the unit circle in the complex plane. Here, the roots of unity help us understand the behavior and distribution of the solutions.
" "Why is it important to find the principal root?
" "The principal root is the primary solution for a given argument. It provides a starting point for understanding the behavior of the other roots and simplifies calculations. In the context of complex numbers, the principal root is often the easiest to compute and interpret.
" "Can this equation be solved by other methods?
" "Yes, alternative methods such as the polar form and De Moivre's theorem can also be used. However, the factorization approach simplifies the problem by breaking it down into more manageable parts. Each method offers its own insights and can be useful depending on the specific equation and context.
" "Keywords
" "Keyword 1: Complex Equations
Keyword 2: Quadratic Form
Keyword 3: Polynomials
Note: The solutions to the equation are not uniquely determined; the roots are equally spaced around a circle. Understanding this distribution is key to fully grasping the problem.
" "By following these steps, you can solve complex equations efficiently and gain a deeper understanding of the underlying mathematical principles. This knowledge can be applied to a wide range of problems in mathematics, engineering, and physics.
" "If you have any further questions or need additional clarifications, feel free to reach out. Happy solving!