Find the square root of `(-8 -6i)`.

Explanation

To find the square root of the complex number \(-8 - 6i\), we can follow these steps:

### Step 1: Set up the equation
Let \( z = \sqrt{-8 - 6i} \). We can express \( z \) in the form \( z = x - iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Square both sides
Squaring both sides gives us:
\[ z^2 = -8 - 6i \implies (x - iy)^2 = -8 - 6i \]

### Step 3: Expand the left side
Expanding the left side:
\[ (x - iy)^2 = x^2 - 2xyi - y^2i^2 \]
Since \( i^2 = -1 \), we can rewrite it as:
\[ x^2 - 2xyi + y^2 = x^2 - y^2 - 2xyi \]

### Step 4: Equate real and imaginary parts
Now we equate the real and imaginary parts:
1. Real part: \( x^2 - y^2 = -8 \) (Equation 1)
2. Imaginary part: \( -2xy = -6 \) or \( 2xy = 6 \) (Equation 2)

### Step 5: Solve for \( xy \)
From Equation 2, we can simplify:
\[ xy = 3 \]

### Step 6: Express \( y \) in terms of \( x \)
From \( xy = 3 \), we can express \( y \) as:
\[ y = \frac{3}{x} \]

### Step 7: Substitute \( y \) in Equation 1
Substituting \( y \) into Equation 1:
\[ x^2 - \left(\frac{3}{x}\right)^2 = -8 \]
This simplifies to:
\[ x^2 - \frac{9}{x^2} = -8 \]

### Step 8: Multiply through by \( x^2 \)
To eliminate the fraction, multiply through by \( x^2 \):
\[ x^4 + 8x^2 - 9 = 0 \]

### Step 9: Let \( u = x^2 \)
Let \( u = x^2 \), then we have a quadratic equation:
\[ u^2 + 8u - 9 = 0 \]

### Step 10: Solve the quadratic equation
Using the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ u = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} = \frac{-8 \pm \sqrt{64 + 36}}{2} = \frac{-8 \pm \sqrt{100}}{2} = \frac{-8 \pm 10}{2} \]
This gives us two possible values for \( u \):
1. \( u = 1 \) (since \( -9 \) is not valid)
2. \( u = -9 \) (not valid as \( u = x^2 \) must be non-negative)

### Step 11: Find \( x \)
Since \( u = x^2 = 1 \), we have:
\[ x = \pm 1 \]

### Step 12: Find \( y \)
Using \( xy = 3 \):
1. If \( x = 1 \), then \( y = 3 \).
2. If \( x = -1 \), then \( y = -3 \).

### Step 13: Write the final answers
Thus, the square roots of \(-8 - 6i\) are:
\[ \sqrt{-8 - 6i} = 1 - 3i \quad \text{and} \quad -1 + 3i \]

### Final Answer
The square root of \(-8 - 6i\) is \( \pm (1 - 3i) \).

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