What are the square roots of `-2i`? `( i=sqrt(-1))`

Options

A. `+-(1+i)`
B. `+-(1-i)`
C. `+-1`
D. `+-1`

Answer

To find the square roots of \(-2i\), we can follow these steps:

### Step 1: Assume the form of the square root
Let the square root of \(-2i\) be represented as \(a + bi\), where \(a\) and \(b\) are real numbers.

### Step 2: Set up the equation
We have:
\[ (a + bi)^2 = -2i \]
Expanding the left side:
\[ a^2 + 2abi + (bi)^2 = a^2 + 2abi - b^2 \]
This simplifies to:
\[ (a^2 - b^2) + (2ab)i = -2i \]

### Step 3: Compare real and imaginary parts
From the equation above, we can equate the real and imaginary parts:
1. Real part: \(a^2 - b^2 = 0\)
2. Imaginary part: \(2ab = -2\)

### Step 4: Solve the real part equation
From \(a^2 - b^2 = 0\), we can write:
\[ a^2 = b^2 \]
This implies:
\[ a = b \quad \text{or} \quad a = -b \]

### Step 5: Solve the imaginary part equation
From \(2ab = -2\), we can simplify:
\[ ab = -1 \]

### Step 6: Substitute \(b\) in terms of \(a\)
Using \(b = a\):
\[ a \cdot a = -1 \implies a^2 = -1 \]
This has no real solutions.

Using \(b = -a\):
\[ a \cdot (-a) = -1 \implies -a^2 = -1 \implies a^2 = 1 \]
Thus, we find:
\[ a = 1 \quad \text{or} \quad a = -1 \]

### Step 7: Find corresponding values of \(b\)
If \(a = 1\), then \(b = -1\).
If \(a = -1\), then \(b = 1\).

### Step 8: Write the square roots
The square roots of \(-2i\) are:
1. \(1 - i\)
2. \(-1 + i\)

Thus, the square roots of \(-2i\) are:
\[ \sqrt{-2i} = 1 - i \quad \text{and} \quad -1 + i \]

### Final Answer
The square roots of \(-2i\) are \(1 - i\) and \(-1 + i\).

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