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Multiplying Complex Numbers: (3 + 2i)(2 + 5i)
This article will guide you through the process of multiplying two complex numbers: (3 + 2i) and (2 + 5i).
Understanding Complex Numbers
Complex numbers are numbers of 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).
Multiplication Process
To multiply complex numbers, we use the distributive property (also known as FOIL - First, Outer, Inner, Last) similar to multiplying binomials:
- Multiply the first terms: 3 * 2 = 6
- Multiply the outer terms: 3 * 5i = 15i
- Multiply the inner terms: 2i * 2 = 4i
- Multiply the last terms: 2i * 5i = 10i²
Now we have: 6 + 15i + 4i + 10i²
- Substitute i² with -1: 6 + 15i + 4i + 10(-1)
- Combine real and imaginary terms: (6 - 10) + (15 + 4)i
- Simplify: -4 + 19i
The Result
Therefore, (3 + 2i)(2 + 5i) = -4 + 19i.
Key Points
- Remember that i² = -1.
- Combine the real and imaginary terms to express the final answer in the standard form a + bi.
By following these steps, you can confidently multiply any complex numbers.