Math Problem Statement

Use square roots to solve the equation x^2 = -64 over the complex numbers. Select any solutions that apply.

Solution

Let's solve the given equation step by step:

The equation is:

x2=64x^2 = -64

Step 1: Solve for xx using square roots

To find xx, take the square root of both sides:

x=±64x = \pm\sqrt{-64}

Step 2: Simplify 64\sqrt{-64}

Using the property of square roots with negative numbers, a=ia\sqrt{-a} = i\sqrt{a}, where ii is the imaginary unit, we can rewrite:

64=64i=8i\sqrt{-64} = \sqrt{64} \cdot i = 8i

Thus, the solutions are:

x=±8ix = \pm 8i

Final Answer:

The correct solutions are:

  • x=8ix = 8i
  • x=8ix = -8i

Options that match:

  • A. 8i8i
  • D. 8i-8i

Would you like further clarification or detailed steps on this?

Here are 5 related questions to expand your understanding:

  1. How does the imaginary unit ii relate to negative square roots?
  2. Can you explain the difference between real and complex numbers?
  3. What are some practical applications of complex numbers in physics or engineering?
  4. How do you represent complex solutions on the complex plane?
  5. Why does (64)(-64) not have real square roots?

Tip: Always remember that square roots of negative numbers lead to complex solutions involving ii.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Square Roots
Imaginary Unit

Formulas

\( \sqrt{-a} = i\sqrt{a} \)

Theorems

Properties of Imaginary Numbers

Suitable Grade Level

Grades 9-12