(4+9i)(2−5i)

Jasonbradley

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Jun 16, 2024

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Multiplying Complex Numbers: (4 + 9i)(2 - 5i)

This article will demonstrate how to multiply two complex numbers, specifically (4 + 9i)(2 - 5i).

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1.

Multiplying Complex Numbers

To multiply complex numbers, we use the distributive property, just like multiplying any two binomials.

Here's how to multiply (4 + 9i)(2 - 5i):

1. Distribute:

   • (4 + 9i)(2 - 5i) = 4(2 - 5i) + 9i(2 - 5i)

2. Simplify:

   • 4(2 - 5i) + 9i(2 - 5i) = 8 - 20i + 18i - 45i²

3. Substitute i² with -1:

   • 8 - 20i + 18i - 45i² = 8 - 20i + 18i - 45(-1)

4. Combine Real and Imaginary Terms:

   • 8 - 20i + 18i - 45(-1) = 8 + 45 - 20i + 18i = 53 - 2i

Result

Therefore, the product of (4 + 9i)(2 - 5i) is 53 - 2i.

Key Points

• Complex numbers are multiplied using the distributive property.
• The term i² is always replaced with -1.
• The final answer is expressed in the form a + bi.