Simplifying Complex Expressions
This article will walk you through the process of simplifying the complex expression: (3 – 2i)(5 + 4i) – (3 – 4i)^2.
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of 1.
StepbyStep Simplification
Let's break down the expression stepbystep:

Expanding the first product: (3 – 2i)(5 + 4i) = (3 * 5) + (3 * 4i) + (2i * 5) + (2i * 4i) = 15 + 12i  10i  8i^2

Expanding the second product: (3  4i)^2 = (3  4i)(3  4i) = (3 * 3) + (3 * 4i) + (4i * 3) + (4i * 4i) = 9  12i  12i + 16i^2

Substituting i^2 with 1: Remember that i^2 = 1. Therefore, we can substitute:
 15 + 12i  10i  8i^2 = 15 + 12i  10i + 8
 9  12i  12i + 16i^2 = 9  12i  12i  16

Combining like terms:
 (15 + 8) + (12i  10i) = 23 + 2i
 (9  16) + (12i  12i) = 7  24i

Final Result: Finally, we subtract the two simplified expressions: (23 + 2i)  (7  24i) = 23 + 2i + 7 + 24i = 30 + 26i
Conclusion
Therefore, the simplified form of the complex expression (3 – 2i)(5 + 4i) – (3 – 4i)^2 is 30 + 26i.