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Multiplying Complex Numbers: A Walkthrough of (3 - 4i)(3 + 4i)
This article will guide you through the process of multiplying the complex numbers (3 - 4i) and (3 + 4i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where:
- a and b are real numbers
- i is the imaginary unit, defined as the square root of -1 (i² = -1)
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property (or FOIL method):
- Multiply the first terms: (3)(3) = 9
- Multiply the outer terms: (3)(4i) = 12i
- Multiply the inner terms: (-4i)(3) = -12i
- Multiply the last terms: (-4i)(4i) = -16i²
Simplifying the Result
Now we have: 9 + 12i - 12i - 16i²
Since i² = -1, we can substitute: 9 + 12i - 12i - 16(-1)
Simplifying further: 9 + 12i - 12i + 16
Combining real and imaginary terms: (9 + 16) + (12 - 12)i
Finally, we get: 25
Conclusion
Therefore, (3 - 4i)(3 + 4i) = 25.
Key takeaway: Multiplying a complex number by its conjugate (the same number with the opposite sign of the imaginary part) always results in a real number. This is a useful property in simplifying complex expressions and solving equations.