%5E2.jpeg "(3-2i)^2")
Expanding (3 - 2i)^2
In mathematics, particularly in complex numbers, expanding expressions involving complex numbers often requires careful attention to the rules of multiplication and the properties of imaginary numbers. Let's explore the expansion of (3 - 2i)^2.
Understanding Complex Numbers
Before we delve into the expansion, let's briefly recall some key aspects of complex numbers:
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Complex Number: A complex number is a number 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 (i.e., i² = -1).
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Squaring a Complex Number: When squaring a complex number, we apply the distributive property of multiplication: (a + bi)² = (a + bi)(a + bi).
Expanding (3 - 2i)²
Following the distributive property, we can expand (3 - 2i)² as follows:
(3 - 2i)² = (3 - 2i)(3 - 2i)
Expanding the product:
(3 - 2i)(3 - 2i) = 3(3 - 2i) - 2i(3 - 2i)
Applying the distributive property again:
9 - 6i - 6i + 4i²
Recall that i² = -1. Substituting:
9 - 6i - 6i + 4(-1)
Simplifying:
9 - 6i - 6i - 4
Combining real and imaginary terms:
(9 - 4) + (-6 - 6)i
Therefore, the expanded form of (3 - 2i)² is 5 - 12i.
Conclusion
Expanding (3 - 2i)² involves applying the distributive property of multiplication and utilizing the fundamental property of the imaginary unit (i² = -1). The result of the expansion is a new complex number, 5 - 12i, which represents the squared value of the original complex number.