The square roots of `7+24i` are

Options

A. `pm (3 + 4i)`
B. `pm (3 +-4i)`
C. `pm (4 + 3i)`
D. `pm (4 - 3i)`

Answer

To find the square roots of the complex number \( 7 + 24i \), we will follow these steps:

### Step 1: Assume the square root
Let the square root of \( 7 + 24i \) be represented as \( x + yi \), where \( x \) and \( y \) are real numbers.

### Step 2: Square both sides
Squaring both sides gives us:
\[ (x + yi)^2 = 7 + 24i \]
Expanding the left side:
\[ x^2 + 2xyi - y^2 = 7 + 24i \]
This can be rewritten as:
\[ (x^2 - y^2) + (2xy)i = 7 + 24i \]

### Step 3: Equate real and imaginary parts
From the above equation, we can equate the real and imaginary parts:
1. \( x^2 - y^2 = 7 \) (Equation 1)
2. \( 2xy = 24 \) (Equation 2)

### Step 4: Solve for \( y \) in terms of \( x \)
From Equation 2:
\[ xy = 12 \implies y = \frac{12}{x} \]

### Step 5: Substitute \( y \) in Equation 1
Substituting \( y \) in Equation 1:
\[ x^2 - \left(\frac{12}{x}\right)^2 = 7 \]
This simplifies to:
\[ x^2 - \frac{144}{x^2} = 7 \]
Multiplying through by \( x^2 \) to eliminate the fraction:
\[ x^4 - 144 = 7x^2 \]
Rearranging gives:
\[ x^4 - 7x^2 - 144 = 0 \]

### Step 6: Let \( z = x^2 \)
Let \( z = x^2 \). Then we have:
\[ z^2 - 7z - 144 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot (-144)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{7 \pm \sqrt{49 + 576}}{2} = \frac{7 \pm \sqrt{625}}{2} = \frac{7 \pm 25}{2} \]
This gives us:
\[ z = \frac{32}{2} = 16 \quad \text{or} \quad z = \frac{-18}{2} = -9 \]
Since \( z = x^2 \), we discard \( z = -9 \) and take \( z = 16 \).

### Step 8: Find \( x \)
Thus, \( x^2 = 16 \) implies:
\[ x = 4 \quad \text{or} \quad x = -4 \]

### Step 9: Find \( y \)
Using \( y = \frac{12}{x} \):
- If \( x = 4 \), then \( y = \frac{12}{4} = 3 \).
- If \( x = -4 \), then \( y = \frac{12}{-4} = -3 \).

### Step 10: Write the square roots
Thus, the square roots of \( 7 + 24i \) are:
\[ 4 + 3i \quad \text{and} \quad -4 - 3i \]

### Final Answer
The square roots of \( 7 + 24i \) are \( 4 + 3i \) and \( -4 - 3i \).

---