The square root of `3+4i` is

Options

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

Answer

To find the square root of the complex number \(3 + 4i\), we can express it in the form \(x + yi\), where \(x\) and \(y\) are real numbers. We will follow these steps:

### Step 1: Set up the equation
Assume that the square root of \(3 + 4i\) can be expressed as:
\[ \sqrt{3 + 4i} = x + yi \]
where \(x\) and \(y\) are real numbers.

### Step 2: Square both sides
Squaring both sides gives:
\[ 3 + 4i = (x + yi)^2 \]
Expanding the right-hand side:
\[ (x + yi)^2 = x^2 + 2xyi + (yi)^2 = x^2 + 2xyi - y^2 \]
Thus, we have:
\[ 3 + 4i = (x^2 - y^2) + (2xy)i \]

### Step 3: Equate real and imaginary parts
From the equation above, we can equate the real and imaginary parts:
1. Real part: \(x^2 - y^2 = 3\)
2. Imaginary part: \(2xy = 4\)

### Step 4: Solve for \(xy\)
From the imaginary part, we can simplify:
\[ xy = 2 \]

### Step 5: Substitute \(y\)
We can express \(y\) in terms of \(x\):
\[ y = \frac{2}{x} \]

### Step 6: Substitute \(y\) into the real part equation
Substituting \(y\) into the real part equation:
\[ x^2 - \left(\frac{2}{x}\right)^2 = 3 \]
This simplifies to:
\[ x^2 - \frac{4}{x^2} = 3 \]
Multiplying through by \(x^2\) to eliminate the fraction:
\[ x^4 - 3x^2 - 4 = 0 \]

### Step 7: Let \(t = x^2\)
Let \(t = x^2\), then we have:
\[ t^2 - 3t - 4 = 0 \]

### Step 8: Factor the quadratic equation
Factoring gives:
\[ (t - 4)(t + 1) = 0 \]
Thus, \(t = 4\) or \(t = -1\). Since \(t = x^2\) must be non-negative, we take:
\[ t = 4 \implies x^2 = 4 \implies x = 2 \text{ or } x = -2 \]

### Step 9: Find corresponding \(y\) values
Using \(xy = 2\):
1. If \(x = 2\), then \(y = \frac{2}{2} = 1\).
2. If \(x = -2\), then \(y = \frac{2}{-2} = -1\).

### Step 10: Write the square roots
Thus, the square roots of \(3 + 4i\) are:
\[ 2 + i \quad \text{and} \quad -2 - i \]

### Final Answer
The square root of \(3 + 4i\) is:
\[ \sqrt{3 + 4i} = 2 + i \quad \text{or} \quad -2 - i \]

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