Multiply. $(4-3 i)(-3+5 i)$
Answer (1)
Expand the product using the distributive property: ( 4 − 3 i ) ( − 3 + 5 i ) = 4 ( − 3 ) + 4 ( 5 i ) − 3 i ( − 3 ) − 3 i ( 5 i ) .
Simplify the terms: − 12 + 20 i + 9 i − 15 i 2 .
Substitute i 2 = − 1 : − 12 + 20 i + 9 i + 15 .
Combine real and imaginary parts: 3 + 29 i .
The final answer is 3 + 29 i .
Explanation
Understanding the Problem We are asked to multiply two complex numbers: ( 4 − 3 i ) and ( − 3 + 5 i ) . Our goal is to express the result in the standard form a + bi , where a and b are real numbers. We will use the distributive property (also known as the FOIL method) to expand the product and then simplify, remembering that i 2 = − 1 .
Expanding the Product Apply the distributive property (FOIL method) to expand the product: ( 4 − 3 i ) ( − 3 + 5 i ) = 4 ( − 3 ) + 4 ( 5 i ) − 3 i ( − 3 ) − 3 i ( 5 i ) = − 12 + 20 i + 9 i − 15 i 2
Substituting i^2 = -1 Now, we replace i 2 with − 1 :
− 12 + 20 i + 9 i − 15 ( − 1 ) = − 12 + 20 i + 9 i + 15
Combining Like Terms Next, we combine the real parts and the imaginary parts: ( − 12 + 15 ) + ( 20 + 9 ) i = 3 + 29 i
Final Answer The final result is 3 + 29 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. Imagine you're designing a circuit. The flow of electricity isn't just a simple on-off switch; it has properties like resistance and reactance. Complex numbers allow engineers to represent these properties in a single mathematical expression, making circuit analysis and design much easier. For example, the impedance of a circuit, which is a measure of its opposition to alternating current, is often expressed as a complex number. By multiplying and manipulating these complex impedances, engineers can predict how a circuit will behave, optimize its performance, and ensure it works reliably.
