Find `sqrt (7-24i)`

Answer

To find the square root of the complex number \( \sqrt{7 - 24i} \), we can express it in the form \( a + bi \), where \( a \) and \( b \) are real numbers. Let's go through the steps to find \( a \) and \( b \).

### Step-by-Step Solution:

1. **Assume the Form**:
Let \( \sqrt{7 - 24i} = a + bi \).

2. **Square Both Sides**:
Squaring both sides gives:
\[ 7 - 24i = (a + bi)^2 \]
Expanding the right-hand side:
\[ (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 + 2abi - b^2 \]
Thus, we have:
\[ 7 - 24i = (a^2 - b^2) + (2ab)i \]

3. **Equate Real and Imaginary Parts**:
From the equation above, we can equate the real and imaginary parts:
- Real part: \( a^2 - b^2 = 7 \) (1)
- Imaginary part: \( 2ab = -24 \) (2)

4. **Express \( b \) in Terms of \( a \)**:
From equation (2), we can express \( b \):
\[ b = \frac{-24}{2a} = \frac{-12}{a} \]

5. **Substitute \( b \) into Equation (1)**:
Substitute \( b \) in equation (1):
\[ a^2 - \left(\frac{-12}{a}\right)^2 = 7 \]
This simplifies to:
\[ a^2 - \frac{144}{a^2} = 7 \]
Multiplying through by \( a^2 \) to eliminate the fraction:
\[ a^4 - 7a^2 - 144 = 0 \]

6. **Let \( x = a^2 \)**:
Let \( x = a^2 \), then we have a quadratic equation:
\[ x^2 - 7x - 144 = 0 \]

7. **Use the Quadratic Formula**:
Applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot (-144)}}{2 \cdot 1} \]
\[ x = \frac{7 \pm \sqrt{49 + 576}}{2} \]
\[ x = \frac{7 \pm \sqrt{625}}{2} \]
\[ x = \frac{7 \pm 25}{2} \]

8. **Calculate Values of \( x \)**:
This gives us two possible values:
\[ x = \frac{32}{2} = 16 \quad \text{and} \quad x = \frac{-18}{2} \text{ (not valid since } x = a^2 \text{ must be non-negative)} \]
Thus, \( a^2 = 16 \) implies \( a = 4 \) or \( a = -4 \).

9. **Find \( b \)**:
Using \( a = 4 \):
\[ b = \frac{-12}{4} = -3 \]
Using \( a = -4 \):
\[ b = \frac{-12}{-4} = 3 \]

10. **Final Values**:
Therefore, the square roots of \( 7 - 24i \) are:
\[ 4 - 3i \quad \text{and} \quad -4 + 3i \]

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