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Squaring a Complex Number: (-5 + 3i)²
This article will guide you through squaring the complex number (-5 + 3i)².
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.
Squaring the Complex Number
To square (-5 + 3i)², we simply multiply it by itself:
(-5 + 3i)² = (-5 + 3i) * (-5 + 3i)
We can expand this using the FOIL method (First, Outer, Inner, Last):
- First: (-5) * (-5) = 25
- Outer: (-5) * (3i) = -15i
- Inner: (3i) * (-5) = -15i
- Last: (3i) * (3i) = 9i²
Now, we know that i² = -1, so we can substitute:
25 - 15i - 15i + 9(-1) = 25 - 15i - 15i - 9
Combining like terms:
25 - 9 - 15i - 15i = 16 - 30i
Conclusion
Therefore, the square of (-5 + 3i)² is 16 - 30i. This demonstrates how to perform basic arithmetic operations with complex numbers.