(x-4%2Bi).jpeg "(x-4-i)(x-4+i)")
Expanding and Simplifying (x - 4 - i)(x - 4 + i)
This expression involves complex numbers and can be expanded using the distributive property (often referred to as FOIL).
Expanding the Expression
-
First: Multiply the first terms of each binomial: (x)(x) = x²
-
Outer: Multiply the outer terms of the binomials: (x)(-4 + i) = -4x + xi
-
Inner: Multiply the inner terms of the binomials: (-4 - i)(x) = -4x - xi
-
Last: Multiply the last terms of each binomial: (-4 - i)(-4 + i) = 16 - 4i + 4i - i²
Simplifying the Expression
Now, combine like terms and remember that i² = -1:
x² - 4x + xi - 4x - xi + 16 - i² = x² - 8x + 16 - (-1) = x² - 8x + 17
Key Points
- Complex Conjugates: Notice that (x - 4 - i) and (x - 4 + i) are complex conjugates. When multiplying complex conjugates, the imaginary terms cancel out, resulting in a real number.
- Factoring: The simplified expression, x² - 8x + 17, represents a quadratic equation. This equation has no real roots.
This process demonstrates how to expand and simplify expressions involving complex numbers. The result shows that even when dealing with complex numbers, we can achieve a simplified real-valued expression through careful application of algebraic properties.