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Subtract the following complex numbers:(2 + 2i) - (14 + 16i)A.-12 + 18iB.12 + 18iC.12 - 14iD.-12 - 14iSUBMITarrow_backPREVIOUS

Question

Subtract the following complex numbers:

(2+2i)(14+16i)(2 + 2i) - (14 + 16i)

A. -12 + 18i
B. 12 + 18i
C. 12 - 14i
D. -12 - 14i

SUBMIT
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Solution

Break Down the Problem

  1. Identify the complex numbers involved:
    • First complex number: z1=2+2i z_1 = 2 + 2i
    • Second complex number: z2=14+16i z_2 = 14 + 16i
  2. Determine the operation:
    • We need to perform the subtraction z1z2 z_1 - z_2 .

Relevant Concepts

To subtract complex numbers, we can use the formula: (z1z2)=(a+bi)(c+di)=(ac)+(bd)i (z_1 - z_2) = (a + bi) - (c + di) = (a - c) + (b - d)i where z1=a+bi z_1 = a + bi and z2=c+di z_2 = c + di .

Analysis and Detail

  1. Apply the formula using the provided complex numbers:
    • Real part: 214=12 2 - 14 = -12
    • Imaginary part: 216=14 2 - 16 = -14
  2. Combine the results: z1z2=1214i z_1 - z_2 = -12 - 14i

Verify and Summarize

By evaluating the real and imaginary parts, the subtraction has been verified. The calculations show that the subtraction of the two complex numbers results in 1214i -12 - 14i .

Final Answer

The result of the subtraction (2+2i)(14+16i) (2 + 2i) - (14 + 16i) is 1214i -12 - 14i . The correct option is D. -12 - 14i.

This problem has been solved

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