(x-4-2i).jpeg "(x-4+2i)(x-4-2i)")
Multiplying Complex Conjugates: (x - 4 + 2i)(x - 4 - 2i)
This expression represents the multiplication of two complex numbers, specifically, complex conjugates. Let's break down the process and explore the significance of this operation:
Understanding Complex Conjugates
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. In our example, the two complex numbers are:
- (x - 4 + 2i)
- (x - 4 - 2i)
The real part is (x - 4) for both, while the imaginary parts are +2i and -2i.
Multiplying Complex Conjugates
To multiply complex conjugates, we can use the distributive property (also known as FOIL):
- First: (x - 4) * (x - 4) = x² - 8x + 16
- Outer: (x - 4) * (-2i) = -2ix + 8i
- Inner: (2i) * (x - 4) = 2ix - 8i
- Last: (2i) * (-2i) = -4i²
Notice that the outer and inner terms cancel each other out: -2ix + 8i + 2ix - 8i = 0
This leaves us with:
x² - 8x + 16 - 4i²
Remember that i² = -1. Substituting this:
x² - 8x + 16 - 4(-1) = x² - 8x + 20
Significance of Multiplying Complex Conjugates
Multiplying complex conjugates always results in a real number. This is because the imaginary terms cancel each other out. This property is useful in various mathematical contexts, including:
- Simplifying complex expressions: When working with complex numbers, multiplying by the conjugate can simplify expressions by eliminating imaginary parts.
- Finding the modulus (magnitude) of a complex number: The modulus of a complex number is the distance from the origin to the point representing the number in the complex plane. The modulus of a complex number is the square root of the product of the number and its conjugate.
- Solving quadratic equations: Complex conjugates often appear as roots of quadratic equations with real coefficients.
In summary, multiplying complex conjugates is a straightforward process with important implications for simplifying expressions and working with complex numbers in different mathematical contexts.