Find the square roots of `-48-14i`.

Answer

To find the square roots of the complex number \(-48 - 14i\), we can follow these steps:

### Step 1: Assume the square root
Let \( z = a + bi \), where \( a \) and \( b \) are real numbers. We want to find \( z \) such that:
\[ z^2 = -48 - 14i \]

### Step 2: Expand the square
Expanding \( z^2 \):
\[ (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 + 2abi - b^2 \]
This can be rewritten as:
\[ (a^2 - b^2) + (2ab)i \]

### Step 3: Set real and imaginary parts equal
From the equation \( z^2 = -48 - 14i \), we can equate the real and imaginary parts:
1. \( a^2 - b^2 = -48 \) (1)
2. \( 2ab = -14 \) (2)

### Step 4: Solve for \( b \) in terms of \( a \)
From equation (2):
\[ b = \frac{-14}{2a} = \frac{-7}{a} \]

### Step 5: Substitute \( b \) into equation (1)
Substituting \( b \) into equation (1):
\[ a^2 - \left(\frac{-7}{a}\right)^2 = -48 \]
This simplifies to:
\[ a^2 - \frac{49}{a^2} = -48 \]

### Step 6: Multiply through by \( a^2 \)
To eliminate the fraction, multiply through by \( a^2 \):
\[ a^4 + 48a^2 - 49 = 0 \]

### Step 7: Let \( x = a^2 \)
Let \( x = a^2 \). The equation becomes:
\[ x^2 + 48x - 49 = 0 \]

### Step 8: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-48 \pm \sqrt{48^2 + 4 \cdot 49}}{2} \]
Calculating the discriminant:
\[ 48^2 = 2304, \quad 4 \cdot 49 = 196 \quad \Rightarrow \quad 2304 + 196 = 2500 \]
Thus,
\[ x = \frac{-48 \pm 50}{2} \]

### Step 9: Solve for \( x \)
Calculating the two possible values:
1. \( x = \frac{2}{2} = 1 \)
2. \( x = \frac{-98}{2} = -49 \) (not possible since \( x = a^2 \) must be non-negative)

### Step 10: Find \( a \) and \( b \)
Since \( x = a^2 = 1 \):
\[ a = \pm 1 \]
Substituting \( a \) back to find \( b \):
\[ b = \frac{-7}{a} \quad \Rightarrow \quad b = -7 \text{ (if } a = 1\text{) or } b = 7 \text{ (if } a = -1\text{)} \]

### Step 11: Write the final answers
Thus, the square roots of \(-48 - 14i\) are:
\[ z = 1 - 7i \quad \text{and} \quad z = -1 + 7i \]
So we can express the final answer as:
\[ \sqrt{-48 - 14i} = \pm (1 - 7i) \]