(x-3-5i)(x-3+5i)

2 min read Jun 17, 2024
(x-3-5i)(x-3+5i)

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.

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