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Simplifying Complex Expressions: (2+3i)(4-5i) + (2-3i)(4+5i)
This article will guide you through simplifying the complex expression: (2+3i)(4-5i) + (2-3i)(4+5i).
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 (i² = -1).
Simplifying the Expression
We can simplify the expression by applying the distributive property and remembering that i² = -1.
Step 1: Expand each product
- (2+3i)(4-5i) = 2(4) + 2(-5i) + 3i(4) + 3i(-5i) = 8 - 10i + 12i - 15i²
- (2-3i)(4+5i) = 2(4) + 2(5i) - 3i(4) - 3i(5i) = 8 + 10i - 12i - 15i²
Step 2: Substitute i² with -1
- 8 - 10i + 12i - 15i² = 8 - 10i + 12i + 15
- 8 + 10i - 12i - 15i² = 8 + 10i - 12i + 15
Step 3: Combine like terms
- (8 - 10i + 12i + 15) + (8 + 10i - 12i + 15) = 23 + 2i
Therefore, the simplified form of the expression (2+3i)(4-5i) + (2-3i)(4+5i) is 23 + 2i.
Conclusion
This process demonstrates how to simplify expressions involving complex numbers by applying the distributive property and substituting i² with -1. It is essential to remember these fundamental concepts to manipulate complex numbers effectively in various mathematical contexts.