Pregunta
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Find all complex number solutions to the equation.

A.
B.
C.
D.
Correct solution set below.

Ask by Malone Chambers. in the United States
Nov 22,2024

Solución de inteligencia artificial de Upstudy

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The correct solution set is B.

Solución

Solve the equation .
Solve the equation by following steps:
  • step0: Solve for :
  • step1: Move the constant to the right side:
  • step2: Remove 0:
  • step3: Simplify the equation:
  • step4: Calculate:
  • step5: Evaluate:

    The complex number solutions to the equation are .
Therefore, the correct solution set is B.

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Bonus Knowledge

To solve , we first rewrite it as . Recognizing that , we find the principal root as . The roots of a cubic equation can be expressed using polar coordinates, leading us to find the complex roots:
  1. The principal root (angle ):
  2. Adding and for the other two roots results in and .
Thus, the complete solution set is represented by option B: .
Finding cube roots in complex analysis can feel like a magical journey into the world of angles and distances! Each root corresponds to a unique point on the complex plane, creating a trio of symmetrical points around the origin. Make sure to visualize these in the argand plane—it’s like plotting a triangle with one vertex at the origin!
Another fun fact: solving equations like these isn’t just about the numbers. It involves understanding how rotations in the complex plane work, which tie back to polynomials and their properties. This concept has important implications in fields like electrical engineering, where alternating current circuits can be understood using complex numbers.

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