The square roots of `- 2 + 2 sqrt(3)i` are :

Options

A. `pm (1 + sqrt(3)i)`
B. `pm (1 - sqrt(3)i)`
C. `pm (-1 + sqrt(3)i)`
D. None of these

Answer

To find the square roots of the complex number \(-2 + 2\sqrt{3}i\), we can follow these steps:

### Step 1: Assume the square root
Let the square root of the complex number be represented as \(x + yi\), where \(x\) and \(y\) are real numbers.

### Step 2: Square both sides
Squaring both sides gives us:
\[ (x + yi)^2 = -2 + 2\sqrt{3}i \]
Expanding the left-hand side:
\[ x^2 + 2xyi - y^2 = -2 + 2\sqrt{3}i \]
This can be separated into real and imaginary parts:
\[ (x^2 - y^2) + (2xy)i = -2 + 2\sqrt{3}i \]

### Step 3: Set up equations
From the real and imaginary parts, we can set up the following equations:
1. \(x^2 - y^2 = -2\) (Equation 1)
2. \(2xy = 2\sqrt{3}\) (Equation 2)

### Step 4: Simplify Equation 2
From Equation 2, we can simplify:
\[ xy = \sqrt{3} \]
This gives us \(y = \frac{\sqrt{3}}{x}\).

### Step 5: Substitute \(y\) in Equation 1
Substituting \(y\) into Equation 1:
\[ x^2 - \left(\frac{\sqrt{3}}{x}\right)^2 = -2 \]
This simplifies to:
\[ x^2 - \frac{3}{x^2} = -2 \]
Multiplying through by \(x^2\) to eliminate the fraction:
\[ x^4 + 2x^2 - 3 = 0 \]

### Step 6: Let \(t = x^2\)
Let \(t = x^2\). Then we have a quadratic equation:
\[ t^2 + 2t - 3 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ t = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2} \]
This gives us two solutions:
\[ t = 1 \quad \text{or} \quad t = -3 \]
Since \(t = x^2\) must be non-negative, we take \(t = 1\).

### Step 8: Find \(x\) and \(y\)
Thus, \(x^2 = 1\) implies \(x = \pm 1\).
Now substituting \(x\) back to find \(y\):
\[ y = \frac{\sqrt{3}}{x} \]
If \(x = 1\), then \(y = \sqrt{3}\).
If \(x = -1\), then \(y = -\sqrt{3}\).

### Step 9: Write the final answers
Thus, the square roots of \(-2 + 2\sqrt{3}i\) are:
\[ 1 + \sqrt{3}i \quad \text{and} \quad -1 - \sqrt{3}i \]

### Final Answer
The square roots of \(-2 + 2\sqrt{3}i\) are:
\[ \pm (1 + \sqrt{3}i) \]