What is one of the square roots of `3+4i,` where `i=sqrt(-1)` ?

Options

A. `2+i`
B. `2-i`
C. `-2+i`
D. `-3-i`

Explanation

To find one of the square roots of the complex number \(3 + 4i\), we can use the following method:

### Step 1: Assume the square root
Let us assume that one of the square roots of \(3 + 4i\) is of the form \(x + yi\), where \(x\) and \(y\) are real numbers.

### Step 2: Set up the equation
We have:
\[ (x + yi)^2 = 3 + 4i \]

### Step 3: Expand the left-hand side
Expanding the left side, we get:
\[ x^2 + 2xyi + (yi)^2 = x^2 + 2xyi - y^2 \]
This can be rearranged as:
\[ (x^2 - y^2) + (2xy)i \]

### Step 4: Equate real and imaginary parts
Now, we equate the real and imaginary parts from both sides:
1. Real part: \(x^2 - y^2 = 3\)
2. Imaginary part: \(2xy = 4\)

### Step 5: Solve the equations
From the imaginary part equation, we can solve for \(y\):
\[ 2xy = 4 \implies xy = 2 \implies y = \frac{2}{x} \quad (1) \]

Now, substitute \(y\) from equation (1) into the real part equation:
\[ x^2 - \left(\frac{2}{x}\right)^2 = 3 \]
This simplifies to:
\[ x^2 - \frac{4}{x^2} = 3 \]

### Step 6: Multiply through by \(x^2\)
To eliminate the fraction, multiply through by \(x^2\):
\[ x^4 - 4 = 3x^2 \]
Rearranging gives us:
\[ x^4 - 3x^2 - 4 = 0 \]

### Step 7: Let \(u = x^2\)
Let \(u = x^2\), then we have:
\[ u^2 - 3u - 4 = 0 \]
This is a quadratic equation in \(u\).

### Step 8: Solve the quadratic equation
Using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ u = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2} \]
This gives us two solutions:
\[ u = \frac{8}{2} = 4 \quad \text{and} \quad u = \frac{-2}{2} = -1 \]
Since \(u = x^2\) must be non-negative, we take \(u = 4\).

### Step 9: Find \(x\)
Thus, \(x^2 = 4\) implies:
\[ x = 2 \quad \text{or} \quad x = -2 \]

### Step 10: Find \(y\)
Using \(y = \frac{2}{x}\):
1. If \(x = 2\), then \(y = \frac{2}{2} = 1\).
2. If \(x = -2\), then \(y = \frac{2}{-2} = -1\).

### Step 11: Write the square roots
Thus, the two square roots of \(3 + 4i\) are:
\[ 2 + i \quad \text{and} \quad -2 - i \]

### Conclusion
One of the square roots of \(3 + 4i\) is:
\[ \boxed{2 + i} \]