find the square roots of the following:
(i) `7+24i`
(ii) `7-24i`
(iii) `-7-24i`
(iv) `-7+24i`.

Answer

To find the square roots of the complex numbers given in the question, we will use the formula for the square root of a complex number \( z = a + bi \). The square root can be expressed in the form \( x + yi \), where \( x \) and \( y \) are real numbers. The following equations must hold:

1. \( x^2 - y^2 = a \)
2. \( 2xy = b \)

We will solve for \( x \) and \( y \) using these equations.

### (i) Finding the square root of \( 7 + 24i \)

1. **Identify \( a \) and \( b \)**: Here, \( a = 7 \) and \( b = 24 \).
2. **Set up the equations**:
- \( x^2 - y^2 = 7 \) (Equation 1)
- \( 2xy = 24 \) (Equation 2)
3. **From Equation 2, solve for \( y \)**:
\[ y = \frac{24}{2x} = \frac{12}{x} \]
4. **Substitute \( y \) into Equation 1**:
\[ x^2 - \left(\frac{12}{x}\right)^2 = 7 \]
\[ x^2 - \frac{144}{x^2} = 7 \]
Multiply through by \( x^2 \):
\[ x^4 - 7x^2 - 144 = 0 \]
5. **Let \( u = x^2 \)**, then we have:
\[ u^2 - 7u - 144 = 0 \]
6. **Use the quadratic formula**:
\[ u = \frac{7 \pm \sqrt{49 + 576}}{2} = \frac{7 \pm 25}{2} \]
This gives:
\[ u = 16 \quad \text{or} \quad u = -9 \]
Since \( u = x^2 \), we take \( u = 16 \) (as \( u = -9 \) is not valid).
7. **Find \( x \)**:
\[ x = 4 \quad \text{or} \quad x = -4 \]
8. **Find \( y \)** using \( y = \frac{12}{x} \):
- If \( x = 4 \): \( y = 3 \)
- If \( x = -4 \): \( y = -3 \)
9. **Final results**:
\[ \sqrt{7 + 24i} = 4 + 3i \quad \text{and} \quad -4 - 3i \]

### (ii) Finding the square root of \( 7 - 24i \)

1. **Identify \( a \) and \( b \)**: Here, \( a = 7 \) and \( b = -24 \).
2. **Set up the equations**:
- \( x^2 - y^2 = 7 \) (Equation 1)
- \( 2xy = -24 \) (Equation 2)
3. **From Equation 2, solve for \( y \)**:
\[ y = \frac{-24}{2x} = \frac{-12}{x} \]
4. **Substitute \( y \) into Equation 1**:
\[ x^2 - \left(\frac{-12}{x}\right)^2 = 7 \]
\[ x^2 - \frac{144}{x^2} = 7 \]
Multiply through by \( x^2 \):
\[ x^4 - 7x^2 - 144 = 0 \]
5. **Let \( u = x^2 \)**, then we have:
\[ u^2 - 7u - 144 = 0 \]
6. **Use the quadratic formula**:
\[ u = \frac{7 \pm 25}{2} \]
This gives:
\[ u = 16 \quad \text{or} \quad u = -9 \]
Since \( u = x^2 \), we take \( u = 16 \).
7. **Find \( x \)**:
\[ x = 4 \quad \text{or} \quad x = -4 \]
8. **Find \( y \)** using \( y = \frac{-12}{x} \):
- If \( x = 4 \): \( y = -3 \)
- If \( x = -4 \): \( y = 3 \)
9. **Final results**:
\[ \sqrt{7 - 24i} = 4 - 3i \quad \text{and} \quad -4 + 3i \]

### (iii) Finding the square root of \( -7 - 24i \)

1. **Identify \( a \) and \( b \)**: Here, \( a = -7 \) and \( b = -24 \).
2. **Set up the equations**:
- \( x^2 - y^2 = -7 \) (Equation 1)
- \( 2xy = -24 \) (Equation 2)
3. **From Equation 2, solve for \( y \)**:
\[ y = \frac{-24}{2x} = \frac{-12}{x} \]
4. **Substitute \( y \) into Equation 1**:
\[ x^2 - \left(\frac{-12}{x}\right)^2 = -7 \]
\[ x^2 - \frac{144}{x^2} = -7 \]
Multiply through by \( x^2 \):
\[ x^4 + 7x^2 - 144 = 0 \]
5. **Let \( u = x^2 \)**, then we have:
\[ u^2 + 7u - 144 = 0 \]
6. **Use the quadratic formula**:
\[ u = \frac{-7 \pm \sqrt{49 + 576}}{2} = \frac{-7 \pm 25}{2} \]
This gives:
\[ u = 9 \quad \text{or} \quad u = -16 \]
Since \( u = x^2 \), we take \( u = 9 \).
7. **Find \( x \)**:
\[ x = 3 \quad \text{or} \quad x = -3 \]
8. **Find \( y \)** using \( y = \frac{-12}{x} \):
- If \( x = 3 \): \( y = -4 \)
- If \( x = -3 \): \( y = 4 \)
9. **Final results**:
\[ \sqrt{-7 - 24i} = 3 - 4i \quad \text{and} \quad -3 + 4i \]

### (iv) Finding the square root of \( -7 + 24i \)

1. **Identify \( a \) and \( b \)**: Here, \( a = -7 \) and \( b = 24 \).
2. **Set up the equations**:
- \( x^2 - y^2 = -7 \) (Equation 1)
- \( 2xy = 24 \) (Equation 2)
3. **From Equation 2, solve for \( y \)**:
\[ y = \frac{24}{2x} = \frac{12}{x} \]
4. **Substitute \( y \) into Equation 1**:
\[ x^2 - \left(\frac{12}{x}\right)^2 = -7 \]
\[ x^2 - \frac{144}{x^2} = -7 \]
Multiply through by \( x^2 \):
\[ x^4 + 7x^2 - 144 = 0 \]
5. **Let \( u = x^2 \)**, then we have:
\[ u^2 + 7u - 144 = 0 \]
6. **Use the quadratic formula**:
\[ u = \frac{-7 \pm \sqrt{49 + 576}}{2} = \frac{-7 \pm 25}{2} \]
This gives:
\[ u = 9 \quad \text{or} \quad u = -16 \]
Since \( u = x^2 \), we take \( u = 9 \).
7. **Find \( x \)**:
\[ x = 3 \quad \text{or} \quad x = -3 \]
8. **Find \( y \)** using \( y = \frac{12}{x} \):
- If \( x = 3 \): \( y = 4 \)
- If \( x = -3 \): \( y = -4 \)
9. **Final results**:
\[ \sqrt{-7 + 24i} = 3 + 4i \quad \text{and} \quad -3 - 4i \]

### Summary of Results
- \( \sqrt{7 + 24i} = 4 + 3i \) and \( -4 - 3i \)
- \( \sqrt{7 - 24i} = 4 - 3i \) and \( -4 + 3i \)
- \( \sqrt{-7 - 24i} = 3 - 4i \) and \( -3 + 4i \)
- \( \sqrt{-7 + 24i} = 3 + 4i \) and \( -3 - 4i \)