(x-3i).jpeg "(x+3i)(x-3i)")
Expanding and Simplifying (x + 3i)(x - 3i)
This expression involves complex numbers, where 'i' represents the imaginary unit (√-1). Let's break down how to expand and simplify it:
Expanding the Expression
We can expand the expression using the FOIL method (First, Outer, Inner, Last) or by recognizing it as a difference of squares:
FOIL Method:
- First: x * x = x²
- Outer: x * -3i = -3xi
- Inner: 3i * x = 3xi
- Last: 3i * -3i = -9i²
Combining the terms: x² - 3xi + 3xi - 9i²
Difference of Squares:
(x + 3i)(x - 3i) is in the form of (a + b)(a - b) which simplifies to a² - b²
Therefore, (x + 3i)(x - 3i) = x² - (3i)²
Simplifying the Expression
Now, we need to simplify the expression further by substituting i² with -1:
FOIL Method:
x² - 3xi + 3xi - 9i² = x² - 9(-1) = x² + 9
Difference of Squares:
x² - (3i)² = x² - 9(i²) = x² - 9(-1) = x² + 9
Conclusion
Therefore, expanding and simplifying (x + 3i)(x - 3i) results in x² + 9. This demonstrates that the product of a complex number and its conjugate (the complex number with the opposite sign for the imaginary part) always results in a real number.