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Question

Difficulty: MediumComplex Numbers and Operations

For the imaginary unit ii, where i2=−1i^2 = -1, what is the value of the expression (3+2i)2−(3−2i)2(3 + 2i)^2 - (3 - 2i)^2?

  1. A
    00
  2. B
    1010
  3. C
    2424
  4. 24i24iAnswer
  5. E
    10+24i10 + 24i

Answer

The correct answer is 24i24i.
The expression can be simplified by expanding each binomial term first. The first term, (3+2i)2(3 + 2i)^2, expands to 9+12i+4i29 + 12i + 4i^2. Since i2=−1i^2 = -1, this simplifies to 9+12i−4=5+12i9 + 12i - 4 = 5 + 12i. The second term, (3−2i)2(3 - 2i)^2, expands to 9−12i+4i29 - 12i + 4i^2, which simplifies to 9−12i−4=5−12i9 - 12i - 4 = 5 - 12i. Subtracting the second simplified term from the first gives (5+12i)−(5−12i)=5−5+12i−(−12i)=24i(5 + 12i) - (5 - 12i) = 5 - 5 + 12i - (-12i) = 24i.

Step-by-Step Solution

1
Expand the first squared binomial expression, (3+2i)2(3 + 2i)^2.
(3+2i)2=9+12i+4i2=9+12i−4=5+12i(3 + 2i)^2 = 9 + 12i + 4i^2 = 9 + 12i - 4 = 5 + 12i
Apply the binomial squaring formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 and use the property of the imaginary unit where i2=−1i^2 = -1.
2
Expand the second squared binomial expression, (3−2i)2(3 - 2i)^2.
(3−2i)2=9−12i+4i2=9−12i−4=5−12i(3 - 2i)^2 = 9 - 12i + 4i^2 = 9 - 12i - 4 = 5 - 12i
Apply the binomial squaring formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2 and use the property of the imaginary unit where i2=−1i^2 = -1.
3
Subtract the second expanded expression from the first.
(5+12i)−(5−12i)=5+12i−5+12i=24i(5 + 12i) - (5 - 12i) = 5 + 12i - 5 + 12i = 24i
Distribute the negative sign to both terms of the second complex number and combine the real and imaginary parts.

Key Concept

Complex Numbers and Operations
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Topics

Complex Numbers and Operations

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