(3-2i).jpeg "(2+i)(3-2i)")
Multiplying Complex Numbers: (2 + i)(3 - 2i)
This article will guide you through multiplying complex numbers, specifically the expression (2 + i)(3 - 2i).
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 (i.e., i² = -1).
Multiplication Process
To multiply complex numbers, we use the distributive property, just like we do with regular binomials. Here's how it works:
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Distribute: Multiply each term in the first complex number by each term in the second complex number.
(2 + i)(3 - 2i) = (2 * 3) + (2 * -2i) + (i * 3) + (i * -2i)
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Simplify: Combine like terms and remember that i² = -1.
= 6 - 4i + 3i - 2i² = 6 - i - 2(-1) = 6 - i + 2
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Combine real and imaginary parts:
= 8 - i
Result
Therefore, the product of (2 + i)(3 - 2i) is 8 - i.
Conclusion
This example illustrates how to multiply complex numbers. By applying the distributive property and the fundamental property of the imaginary unit (i² = -1), we can simplify complex expressions and obtain a result in the standard form a + bi.