Multiply: $(-4+i)(2-3 i)$
Answer (1)
Multiply the complex numbers using the distributive property: ( − 4 + i ) ( 2 − 3 i ) = − 4 ( 2 ) + ( − 4 ) ( − 3 i ) + i ( 2 ) + i ( − 3 i ) .
Simplify the expression: − 8 + 12 i + 2 i − 3 i 2 .
Substitute i 2 = − 1 : − 8 + 12 i + 2 i + 3 .
Combine real and imaginary parts to get the final answer: − 5 + 14 i .
Explanation
Understanding the Problem We are asked to multiply two complex numbers: ( − 4 + i ) and ( 2 − 3 i ) . The result should be in the form a + bi , where a and b are real numbers.
Applying the Distributive Property To multiply the two complex numbers, we use the distributive property (also known as the FOIL method). This means we multiply each term in the first complex number by each term in the second complex number:
( − 4 + i ) ( 2 − 3 i ) = − 4 ( 2 ) + ( − 4 ) ( − 3 i ) + i ( 2 ) + i ( − 3 i )
Simplifying the Terms Now, let's simplify each term:
( − 4 ) ( 2 ) = − 8 ( − 4 ) ( − 3 i ) = 12 i ( i ) ( 2 ) = 2 i ( i ) ( − 3 i ) = − 3 i 2
So, we have: − 8 + 12 i + 2 i − 3 i 2
Substituting i 2 = − 1 Recall that i 2 = − 1 . Substitute this into the expression:
− 8 + 12 i + 2 i − 3 ( − 1 ) = − 8 + 12 i + 2 i + 3
Combining Real and Imaginary Parts Now, combine the real parts and the imaginary parts:
Real parts: − 8 + 3 = − 5 Imaginary parts: 12 i + 2 i = 14 i
So, the result is: − 5 + 14 i
Final Answer The product of the two complex numbers ( − 4 + i ) ( 2 − 3 i ) is − 5 + 14 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. Imagine designing a circuit where you need to analyze alternating current (AC). Complex numbers help represent the impedance (resistance) in these circuits, making calculations much easier. For example, if you have a circuit element with a resistance of 4 ohms and a reactance of 3 ohms, you can represent the impedance as the complex number 4 + 3 i . Multiplying or dividing these complex impedances helps engineers understand how the circuit will behave, ensuring efficient and stable performance. This is just one of many applications where complex numbers simplify real-world problems!
