Math Problem Statement

solve

Use square roots to solve the equation x² = -25 over the complex numbers. Select all solutions for the equation.

Solution

To solve the equation x2=25x^2 = -25 over the complex numbers, we follow these steps:

  1. Rewrite the equation: x2=25x^2 = -25

  2. Take the square root of both sides: x=±25x = \pm \sqrt{-25}

  3. Recall that 25\sqrt{-25} can be expressed as 251\sqrt{25} \cdot \sqrt{-1}. Using i=1i = \sqrt{-1}, we get: 25=5i\sqrt{-25} = 5i

  4. Therefore, the solutions are: x=±5ix = \pm 5i

The two solutions are: x=5iandx=5ix = 5i \quad \text{and} \quad x = -5i

Correct Answers:

  • 5i5i
  • 5i-5i

Let me know if you'd like further explanation or have any questions. Here are 5 related questions to expand your understanding:

  1. What is the general process for solving equations involving complex numbers?
  2. How do square roots of negative numbers work in terms of imaginary numbers?
  3. Why is i2=1i^2 = -1 an essential property in complex numbers?
  4. Can you solve similar equations like x2=49x^2 = -49 or x2=4x^2 = -4?
  5. What is the geometric interpretation of solutions like ±5i\pm 5i on the complex plane?

Tip: When solving equations involving complex numbers, always express negative square roots in terms of ii, and check for both positive and negative solutions!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Square Roots
Imaginary Numbers

Formulas

x² = -a implies x = ±√(-a)
√(-a) = √a * i where i² = -1

Theorems

Properties of Square Roots
Definition of Imaginary Unit (i² = -1)

Suitable Grade Level

Grades 9-12