The square root of the numbers `(- 7 - 24i)` is

Options

A. (3 +4i)
B. (3-4i)
C. `pm (3 - 4i)`
D. none

Answer

To find the square root of the complex number \(-7 - 24i\), we can follow these steps:

### Step 1: Set up the equation
Let \( z = x + yi \) be the square root of \(-7 - 24i\). Then we have:
\[ z^2 = -7 - 24i \]
This implies:
\[ (x + yi)^2 = -7 - 24i \]

### Step 2: Expand the left side
Expanding the left side gives:
\[ x^2 + 2xyi - y^2 = -7 - 24i \]
This can be separated into real and imaginary parts:
\[ (x^2 - y^2) + (2xy)i = -7 - 24i \]

### Step 3: Set up the equations
From the real and imaginary parts, we can set up the following equations:
1. \( x^2 - y^2 = -7 \) (Equation 1)
2. \( 2xy = -24 \) (Equation 2)

### Step 4: Solve for \(xy\)
From Equation 2, we can express \(xy\):
\[ xy = -12 \]

### Step 5: Substitute \(y\) in terms of \(x\)
From \(xy = -12\), we can express \(y\) as:
\[ y = \frac{-12}{x} \]

### Step 6: Substitute \(y\) into Equation 1
Substituting \(y\) into Equation 1 gives:
\[ x^2 - \left(\frac{-12}{x}\right)^2 = -7 \]
This simplifies to:
\[ x^2 - \frac{144}{x^2} = -7 \]

### Step 7: Multiply through by \(x^2\)
To eliminate the fraction, multiply through by \(x^2\):
\[ x^4 + 7x^2 - 144 = 0 \]
Let \(u = x^2\), then we have:
\[ u^2 + 7u - 144 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ u = \frac{-7 \pm \sqrt{7^2 + 4 \cdot 144}}{2} \]
\[ u = \frac{-7 \pm \sqrt{49 + 576}}{2} \]
\[ u = \frac{-7 \pm \sqrt{625}}{2} \]
\[ u = \frac{-7 \pm 25}{2} \]
Calculating the two possible values:
1. \(u = \frac{18}{2} = 9\)
2. \(u = \frac{-32}{2} = -16\) (not valid since \(u = x^2\) must be non-negative)

Thus, \(u = 9\), which means:
\[ x^2 = 9 \implies x = 3 \text{ or } x = -3 \]

### Step 9: Find corresponding \(y\) values
Using \(xy = -12\):
1. If \(x = 3\), then \(3y = -12 \implies y = -4\).
2. If \(x = -3\), then \(-3y = -12 \implies y = 4\).

### Step 10: Write the square roots
Thus, the two square roots of \(-7 - 24i\) are:
1. \(z_1 = 3 - 4i\)
2. \(z_2 = -3 + 4i\)

### Final Answer
The square roots of the complex number \(-7 - 24i\) are:
\[ 3 - 4i \quad \text{and} \quad -3 + 4i \]