(x-5%2B2i).jpeg "(x-5-2i)(x-5+2i)")
Expanding and Simplifying (x-5-2i)(x-5+2i)
This expression involves complex numbers and we will use the distributive property (also known as FOIL) to expand and simplify it.
Step 1: Expanding using FOIL
FOIL stands for First, Outer, Inner, Last. We will multiply each term in the first binomial with each term in the second binomial.
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First: (x)(x) = x²
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Outer: (x)(2i) = 2ix
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Inner: (-5)(x) = -5x
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Last: (-5)(2i) = -10i
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First: (x)(-5) = -5x
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Outer: (x)(-2i) = -2ix
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Inner: (-5)(-5) = 25
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Last: (-5)(-2i) = 10i
Step 2: Combining Like Terms
Now, we combine all the terms:
x² + 2ix - 5x - 10i - 5x - 2ix + 25 + 10i
Notice that the terms with 'i' cancel each other out (2ix - 2ix) and (-10i + 10i).
Step 3: Simplifying the Expression
The final simplified expression is:
x² - 10x + 25
Important Observation:
The final result is a quadratic expression. This is because we multiplied two complex conjugates together. Complex conjugates are complex numbers with the same real part but opposite imaginary parts. The product of complex conjugates always results in a real number.