Simplifying (x+3i)(x3i) into Standard Form
This expression represents the product of two complex conjugates. Let's break down the process of simplifying it into standard form, which is a + bi, where a and b are real numbers, and i is the imaginary unit (√1).
Understanding Complex Conjugates
Complex conjugates are pairs of complex numbers that differ only in the sign of their imaginary parts. In this case, we have:
 (x + 3i) : This is the first complex number.
 (x  3i) : This is the complex conjugate of the first number.
Simplifying the Expression
To simplify the product, we can use the difference of squares pattern: (a + b)(a  b) = a²  b²
Applying this pattern to our expression:

Identify a and b:
 a = x
 b = 3i

Substitute into the pattern: (x + 3i)(x  3i) = x²  (3i)²

Simplify: x²  (3i)² = x²  9i²

Remember that i² = 1: x²  9i² = x²  9(1)

Final simplification: x²  9(1) = x² + 9
Conclusion
Therefore, the simplified form of (x+3i)(x3i) in standard form is x² + 9. This result demonstrates that the product of complex conjugates always results in a real number.