(x-4i)(x+4i)

Jasonbradley

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Jun 17, 2024

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Multiplying Complex Numbers: (x - 4i)(x + 4i)

This problem involves multiplying two complex numbers in the form of (a + bi)(a - bi), which is a special case known as the difference of squares.

Understanding the Difference of Squares

The difference of squares pattern states that: (a + b)(a - b) = a² - b²

Applying the Pattern

Let's apply this pattern to our problem:

• a = x
• b = 4i

Substituting these values into the difference of squares pattern:

(x + 4i)(x - 4i) = x² - (4i)²

Simplifying the Expression

Remember that i² = -1. Therefore:

• x² - (4i)² = x² - (4² * i²)
• = x² - (16 * -1)
• = x² + 16

Conclusion

The product of (x - 4i)(x + 4i) simplifies to x² + 16. This demonstrates that multiplying complex numbers in the form of (a + bi)(a - bi) results in a real number, specifically the sum of the squares of the real and imaginary parts.