Which subtraction expression has the difference $1+4 i$?
A. $(-2+6 i)-(-1-2 i)$
B. $(3+5 i)-(2+i)$
C. $(3+5 i)-(2-i)$
D. $(-2+6 i)-(1-2 i)$
Answer (2)
Evaluate the first expression: ( − 2 + 6 i ) − ( − 1 − 2 i ) = − 1 + 8 i .
Evaluate the second expression: ( 3 + 5 i ) − ( 2 + i ) = 1 + 4 i .
Evaluate the third expression: ( 3 + 5 i ) − ( 2 − i ) = 1 + 6 i .
Evaluate the fourth expression: ( − 2 + 6 i ) − ( 1 − 2 i ) = − 3 + 8 i . The subtraction expression that has the difference 1 + 4 i is ( 3 + 5 i ) − ( 2 + i ) .
Explanation
Problem Analysis We are asked to find which subtraction expression results in the complex number 1 + 4 i . We will evaluate each of the given options to determine which one matches the target complex number.
Evaluating the First Expression Let's evaluate the first expression: ( − 2 + 6 i ) − ( − 1 − 2 i ) . This simplifies to − 2 + 6 i + 1 + 2 i . Combining the real parts, we have − 2 + 1 = − 1 . Combining the imaginary parts, we have 6 i + 2 i = 8 i . Thus, the result is − 1 + 8 i .
Evaluating the Second Expression Now, let's evaluate the second expression: ( 3 + 5 i ) − ( 2 + i ) . This simplifies to 3 + 5 i − 2 − i . Combining the real parts, we have 3 − 2 = 1 . Combining the imaginary parts, we have 5 i − i = 4 i . Thus, the result is 1 + 4 i .
Evaluating the Third Expression Next, let's evaluate the third expression: ( 3 + 5 i ) − ( 2 − i ) . This simplifies to 3 + 5 i − 2 + i . Combining the real parts, we have 3 − 2 = 1 . Combining the imaginary parts, we have 5 i + i = 6 i . Thus, the result is 1 + 6 i .
Evaluating the Fourth Expression Finally, let's evaluate the fourth expression: ( − 2 + 6 i ) − ( 1 − 2 i ) . This simplifies to − 2 + 6 i − 1 + 2 i . Combining the real parts, we have − 2 − 1 = − 3 . Combining the imaginary parts, we have 6 i + 2 i = 8 i . Thus, the result is − 3 + 8 i .
Identifying the Correct Expression Comparing the results, we see that the second expression, ( 3 + 5 i ) − ( 2 + i ) , yields the complex number 1 + 4 i , which is the target complex number.
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. The voltage and current in an AC circuit can be represented as complex numbers, and the impedance of the circuit, which is the opposition to the flow of current, is also a complex number. By using complex numbers, engineers can analyze and design AC circuits more easily.
The subtraction expression that yields the difference 1 + 4 i is ( 3 + 5 i ) − ( 2 + i ) . After evaluating each option, only the second expression produced the desired result.
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