(x-4+2i)(x-4-2i)

3 min read Jun 17, 2024
(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):

  1. First: (x - 4) * (x - 4) = x² - 8x + 16
  2. Outer: (x - 4) * (-2i) = -2ix + 8i
  3. Inner: (2i) * (x - 4) = 2ix - 8i
  4. 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.

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