The square root of the number `5 + 12 i` is

Options

A. `(3 +2i)`
B. `(3 - 2i)`
C. `pm (3 +2i)`
D. none

Answer

To find the square root of the complex number \( 5 + 12i \), we can follow these steps:

### Step 1: Assume the square root
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then, we have:
\[ z^2 = 5 + 12i \]

### Step 2: Expand \( z^2 \)
Expanding \( z^2 \) gives:
\[ (x + yi)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + (2xy)i \]
Thus, we can equate the real and imaginary parts:
\[ x^2 - y^2 = 5 \quad \text{(1)} \]
\[ 2xy = 12 \quad \text{(2)} \]

### Step 3: Solve for \( xy \)
From equation (2), we can express \( xy \) as:
\[ xy = 6 \quad \text{(3)} \]

### Step 4: Substitute \( y \) in terms of \( x \)
From equation (3), we can express \( y \) in terms of \( x \):
\[ y = \frac{6}{x} \]

### Step 5: Substitute \( y \) into equation (1)
Substituting \( y \) into equation (1):
\[ x^2 - \left(\frac{6}{x}\right)^2 = 5 \]
This simplifies to:
\[ x^2 - \frac{36}{x^2} = 5 \]
Multiplying through by \( x^2 \) to eliminate the fraction:
\[ x^4 - 5x^2 - 36 = 0 \]

### Step 6: Let \( u = x^2 \)
Let \( u = x^2 \). The equation becomes:
\[ u^2 - 5u - 36 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ u = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-36)}}{2 \cdot 1} \]
\[ u = \frac{5 \pm \sqrt{25 + 144}}{2} \]
\[ u = \frac{5 \pm \sqrt{169}}{2} \]
\[ u = \frac{5 \pm 13}{2} \]
Calculating the two possible values:
\[ u = \frac{18}{2} = 9 \quad \text{or} \quad u = \frac{-8}{2} = -4 \]
Since \( u = x^2 \) must be non-negative, we take \( u = 9 \).

### Step 8: Find \( x \)
Thus, we have:
\[ x^2 = 9 \implies x = 3 \text{ or } x = -3 \]

### Step 9: Find \( y \) using \( xy = 6 \)
Using \( xy = 6 \):
1. If \( x = 3 \):
\[ 3y = 6 \implies y = 2 \]
2. If \( x = -3 \):
\[ -3y = 6 \implies y = -2 \]

### Step 10: Write the solutions for \( z \)
Thus, we have two possible values for \( z \):
\[ z = 3 + 2i \quad \text{and} \quad z = -3 - 2i \]

### Final Answer
The square roots of \( 5 + 12i \) are:
\[ z = 3 + 2i \quad \text{and} \quad z = -3 - 2i \]

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