Which subtraction expression has the difference [tex]$1+4 i$[/tex]?
A. [tex]$(-2+6 i)-(-1-2 i)$[/tex]
B. [tex]$(3+5 i)-(2+i)$[/tex]
C. [tex]$(3+5 i)-(2-i)$[/tex]
D. [tex]$(-2+6 i)-(1-2 i)$[/tex]
Answer (2)
Evaluate each subtraction expression.
Expression 1: ( − 2 + 6 i ) − ( − 1 − 2 i ) = − 1 + 8 i .
Expression 2: ( 3 + 5 i ) − ( 2 + i ) = 1 + 4 i .
Expression 3: ( 3 + 5 i ) − ( 2 − i ) = 1 + 6 i .
Expression 4: ( − 2 + 6 i ) − ( 1 − 2 i ) = − 3 + 8 i .
The expression with the difference 1 + 4 i is ( 3 + 5 i ) − ( 2 + i ) , so the answer is ( 3 + 5 i ) − ( 2 + i ) .
Explanation
Understanding the Problem We are given four subtraction expressions involving complex numbers and we want to find the expression that results in the complex number 1 + 4 i . We will evaluate each expression separately.
Evaluating Expression 1 Expression 1: ( − 2 + 6 i ) − ( − 1 − 2 i ) = − 2 + 6 i + 1 + 2 i = ( − 2 + 1 ) + ( 6 + 2 ) i = − 1 + 8 i .
Evaluating Expression 2 Expression 2: ( 3 + 5 i ) − ( 2 + i ) = 3 + 5 i − 2 − i = ( 3 − 2 ) + ( 5 − 1 ) i = 1 + 4 i .
Evaluating Expression 3 Expression 3: ( 3 + 5 i ) − ( 2 − i ) = 3 + 5 i − 2 + i = ( 3 − 2 ) + ( 5 + 1 ) i = 1 + 6 i .
Evaluating Expression 4 Expression 4: ( − 2 + 6 i ) − ( 1 − 2 i ) = − 2 + 6 i − 1 + 2 i = ( − 2 − 1 ) + ( 6 + 2 ) i = − 3 + 8 i .
Finding the Correct Expression Comparing the results of each expression with the target difference 1 + 4 i , we find that Expression 2 matches the target difference. Therefore, the subtraction expression that has the difference 1 + 4 i is ( 3 + 5 i ) − ( 2 + i ) .
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. The voltage, current, and impedance in an AC circuit can be represented as complex numbers. By using complex numbers, engineers can analyze and design AC circuits more easily. For example, they can calculate the total impedance of a circuit by adding the complex impedances of individual components.
The subtraction expression that results in the difference 1 + 4 i is Expression B: ( 3 + 5 i ) − ( 2 + i ) . After evaluating each expression, we find that only Expression B produces the desired result. Thus, the correct option is B.
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