Find the square root of the complex number
`-4-3i`

Answer

To find the square root of the complex number \(-4 - 3i\), we can follow these steps:

### Step 1: Set Up the Equation
Let \( z = \sqrt{-4 - 3i} \). We can express \( z \) in terms of real numbers \( a \) and \( b \):
\[ z = a + bi \]
where \( a \) and \( b \) are real numbers.

### Step 2: Square Both Sides
Squaring both sides gives:
\[ z^2 = a^2 + 2abi - b^2 = -4 - 3i \]
This means we can equate the real and imaginary parts:
\[ a^2 - b^2 = -4 \quad \text{(1)} \]
\[ 2ab = -3 \quad \text{(2)} \]

### Step 3: Solve for \( b \)
From equation (2), we can express \( b \) in terms of \( a \):
\[ b = \frac{-3}{2a} \quad \text{(3)} \]

### Step 4: Substitute \( b \) into Equation (1)
Substituting equation (3) into equation (1):
\[ a^2 - \left(\frac{-3}{2a}\right)^2 = -4 \]
This simplifies to:
\[ a^2 - \frac{9}{4a^2} = -4 \]

### Step 5: Clear the Denominator
Multiply through by \( 4a^2 \) to eliminate the fraction:
\[ 4a^4 + 16a^2 - 9 = 0 \]

### Step 6: Let \( x = a^2 \)
Let \( x = a^2 \). The equation becomes:
\[ 4x^2 + 16x - 9 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4} \]
\[ x = \frac{-16 \pm \sqrt{256 + 144}}{8} \]
\[ x = \frac{-16 \pm \sqrt{400}}{8} \]
\[ x = \frac{-16 \pm 20}{8} \]
Calculating the two possible values:
1. \( x = \frac{4}{8} = \frac{1}{2} \)
2. \( x = \frac{-36}{8} \) (not possible since \( x \) must be non-negative)

Thus, \( a^2 = \frac{1}{2} \) implies:
\[ a = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]

### Step 8: Find \( b \)
Substituting \( a \) back into equation (3):
\[ b = \frac{-3}{2a} = \frac{-3}{2 \cdot \frac{\sqrt{2}}{2}} = \frac{-3}{\sqrt{2}} \quad \text{(for } a = \frac{\sqrt{2}}{2}\text{)} \]
And for \( a = -\frac{\sqrt{2}}{2} \):
\[ b = \frac{-3}{-2 \cdot \frac{\sqrt{2}}{2}} = \frac{3}{\sqrt{2}} \]

### Step 9: Write the Square Roots
Thus, we have two possible square roots:
1. \( z_1 = \frac{\sqrt{2}}{2} - \frac{3}{\sqrt{2}}i \)
2. \( z_2 = -\frac{\sqrt{2}}{2} + \frac{3}{\sqrt{2}}i \)

### Final Answer
The square roots of the complex number \(-4 - 3i\) are:
\[ z_1 = \frac{\sqrt{2}}{2} - \frac{3}{\sqrt{2}}i \quad \text{and} \quad z_2 = -\frac{\sqrt{2}}{2} + \frac{3}{\sqrt{2}}i \]

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