(x-3-2i).jpeg "(x-3+2i)(x-3-2i)")
Expanding the Expression (x-3+2i)(x-3-2i)
This expression involves complex numbers and can be expanded using the FOIL method (First, Outer, Inner, Last).
Let's break down the steps:
1. Expanding the expression using FOIL:
- First: (x) * (x) = x²
- Outer: (x) * (-3-2i) = -3x - 2ix
- Inner: (-3+2i) * (x) = -3x + 2ix
- Last: (-3+2i) * (-3-2i) = 9 + 6i - 6i - 4i²
2. Combining like terms:
Notice that the terms -2ix and +2ix cancel each other out. We also know that i² = -1. Combining everything:
x² - 3x - 3x + 9 + 4 = x² - 6x + 13
Therefore, the expanded form of (x-3+2i)(x-3-2i) is x² - 6x + 13.
Important Note:
This result demonstrates a common pattern in complex numbers. When multiplying a complex number by its conjugate (obtained by changing the sign of the imaginary part), the result is always a real number. In this case, the conjugate of (x-3+2i) is (x-3-2i).