Find the square root of the following complex numbers
`3+4i`

Answer

To find the square root of the complex number \(3 + 4i\), we can follow these steps:

### Step 1: Assume the square root
Let the square root of \(3 + 4i\) be of the form:
\[ \sqrt{3 + 4i} = x + yi \]
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 can rewrite the equation as:
\[ 3 + 4i = (x^2 - y^2) + (2xy)i \]

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

### Step 4: Solve for \(y\) in terms of \(x\)
From Equation 2, we can solve for \(y\):
\[ 2xy = 4 \implies xy = 2 \implies y = \frac{2}{x} \]

### Step 5: Substitute \(y\) into Equation 1
Substituting \(y\) into Equation 1:
\[ x^2 - \left(\frac{2}{x}\right)^2 = 3 \]
This simplifies to:
\[ x^2 - \frac{4}{x^2} = 3 \]
Multiplying through by \(x^2\) to eliminate the fraction:
\[ x^4 - 4 = 3x^2 \]
Rearranging gives:
\[ x^4 - 3x^2 - 4 = 0 \]

### Step 6: Let \(u = x^2\)
Let \(u = x^2\). Then we have:
\[ u^2 - 3u - 4 = 0 \]
This is a quadratic equation in \(u\).

### Step 7: Solve the quadratic equation
Using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ u = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \]
\[ u = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm 5}{2} \]
Calculating the two possible values for \(u\):
1. \(u = \frac{8}{2} = 4\)
2. \(u = \frac{-2}{2} = -1\) (not valid since \(u\) must be non-negative)

Thus, \(u = 4\) implies:
\[ x^2 = 4 \implies x = 2 \text{ or } x = -2 \]

### Step 8: 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 9: Write the square roots
Thus, the square roots of \(3 + 4i\) are:
\[ \sqrt{3 + 4i} = 2 + i \quad \text{and} \quad -2 - i \]

### Final Answer
The square roots of the complex number \(3 + 4i\) are:
\[ 2 + i \quad \text{and} \quad -2 - i \]