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Multiplying Complex Numbers: (x - 3 - 5i)(x - 3 + 5i)
This expression involves multiplying two complex numbers. Let's break down the process and see the result:
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where:
- a is the real part.
- b is the imaginary part.
- i is the imaginary unit, where i² = -1.
Expanding the Expression
We can multiply these complex numbers using the distributive property (or FOIL method):
(x - 3 - 5i)(x - 3 + 5i) = x(x - 3 + 5i) - 3(x - 3 + 5i) - 5i(x - 3 + 5i)
Now, let's distribute each term:
= x² - 3x + 5ix - 3x + 9 - 15i - 5ix + 15i - 25i²
Notice that the terms with 'i' cancel each other out:
= x² - 3x - 3x + 9 - 25i²
Simplifying the Expression
Recall that i² = -1, so we can substitute:
= x² - 3x - 3x + 9 - 25(-1)
= x² - 6x + 9 + 25
Finally, we combine like terms:
= x² - 6x + 34
The Result
The product of the complex numbers (x - 3 - 5i) and (x - 3 + 5i) is the real quadratic expression x² - 6x + 34. This demonstrates an important property of complex conjugates: the product of a complex number and its conjugate is always a real number.