Find the square root of the following complex numbers
`-40-42i`

Answer

To find the square root of the complex number \(-40 - 42i\), we can follow these steps:

### Step 1: Assume the square root
Let the square root of \(-40 - 42i\) be \(a + bi\), where \(a\) and \(b\) are real numbers.

### Step 2: Square both sides
Squaring both sides gives us:
\[ (a + bi)^2 = -40 - 42i \]
Expanding the left-hand side:
\[ a^2 + 2abi - b^2 = -40 - 42i \]
This can be rearranged to:
\[ (a^2 - b^2) + (2ab)i = -40 - 42i \]

### Step 3: Equate real and imaginary parts
From this equation, we can equate the real and imaginary parts:
1. \(a^2 - b^2 = -40\) (1)
2. \(2ab = -42\) (2)

### Step 4: Solve for \(ab\)
From equation (2), we can express \(ab\):
\[ ab = -21 \quad \text{(3)} \]

### Step 5: Substitute \(b\) in terms of \(a\)
From equation (3), we can express \(b\) in terms of \(a\):
\[ b = \frac{-21}{a} \quad \text{(4)} \]

### Step 6: Substitute \(b\) in equation (1)
Substituting equation (4) into equation (1):
\[ a^2 - \left(\frac{-21}{a}\right)^2 = -40 \]
This simplifies to:
\[ a^2 - \frac{441}{a^2} = -40 \]

### Step 7: Multiply through by \(a^2\)
To eliminate the fraction, multiply through by \(a^2\):
\[ a^4 + 40a^2 - 441 = 0 \]

### Step 8: Let \(x = a^2\)
Let \(x = a^2\). The equation becomes:
\[ x^2 + 40x - 441 = 0 \]

### Step 9: Solve the quadratic equation
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{-40 \pm \sqrt{40^2 + 4 \cdot 441}}{2} \]
Calculating the discriminant:
\[ x = \frac{-40 \pm \sqrt{1600 + 1764}}{2} = \frac{-40 \pm \sqrt{3364}}{2} \]
\[ \sqrt{3364} = 58 \]
Thus,
\[ x = \frac{-40 \pm 58}{2} \]
Calculating the two possible values:
1. \(x = \frac{18}{2} = 9\)
2. \(x = \frac{-98}{2} = -49\) (not possible since \(x = a^2\) must be non-negative)

### Step 10: Find \(a\)
Since \(x = 9\), we have:
\[ a^2 = 9 \implies a = \pm 3 \]

### Step 11: Find \(b\)
Using equation (4) to find \(b\):
1. If \(a = 3\):
\[ b = \frac{-21}{3} = -7 \]
2. If \(a = -3\):
\[ b = \frac{-21}{-3} = 7 \]

### Step 12: Write the final answer
Thus, the square roots of \(-40 - 42i\) are:
\[ 3 - 7i \quad \text{and} \quad -3 + 7i \]
We can write the final answer as:
\[ \sqrt{-40 - 42i} = \pm (3 - 7i) \]