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Multiplying Complex Numbers: (3 + 4i)(2 - 5i)
This article will guide you through the process of multiplying two complex numbers: (3 + 4i) and (2 - 5i).
Understanding Complex Numbers
Before we dive into the multiplication, let's quickly define what complex numbers are:
- 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 (i² = -1).
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, just like we do with real numbers. This means we multiply each term of the first complex number by each term of the second complex number.
Here's how we multiply (3 + 4i)(2 - 5i):
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Expand using the distributive property: (3 + 4i)(2 - 5i) = 3(2 - 5i) + 4i(2 - 5i)
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Distribute: = 6 - 15i + 8i - 20i²
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Substitute i² with -1: = 6 - 15i + 8i - 20(-1)
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Simplify by combining real and imaginary terms: = 6 + 20 - 15i + 8i
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Combine like terms: = 26 - 7i
Result
Therefore, the product of (3 + 4i) and (2 - 5i) is 26 - 7i.