Find the square root of the complex number
`5+12i`

Answer

To find the square root of the complex number \(5 + 12i\), we can express it in the form \(a + bi\) where \(a\) and \(b\) are real numbers. We will follow these steps:

### Step 1: Set up the equation
Assume that the square root of \(5 + 12i\) can be expressed as:
\[ \sqrt{5 + 12i} = a + bi \]
where \(a\) and \(b\) are real numbers.

### Step 2: Square both sides
Squaring both sides gives us:
\[ (a + bi)^2 = 5 + 12i \]
Expanding the left side, we have:
\[ a^2 + 2abi + (bi)^2 = 5 + 12i \]
Since \(i^2 = -1\), this simplifies to:
\[ a^2 - b^2 + 2abi = 5 + 12i \]

### Step 3: Separate real and imaginary parts
Now, we can separate the real and imaginary parts:
- Real part: \(a^2 - b^2 = 5\) (Equation 1)
- Imaginary part: \(2ab = 12\) (Equation 2)

### Step 4: Solve for \(b\) in terms of \(a\)
From Equation 2, we can solve for \(b\):
\[ b = \frac{12}{2a} = \frac{6}{a} \]

### Step 5: Substitute \(b\) into Equation 1
Substituting \(b\) into Equation 1:
\[ a^2 - \left(\frac{6}{a}\right)^2 = 5 \]
This simplifies to:
\[ a^2 - \frac{36}{a^2} = 5 \]

### Step 6: Multiply through by \(a^2\)
To eliminate the fraction, multiply through by \(a^2\):
\[ a^4 - 36 = 5a^2 \]
Rearranging gives us:
\[ a^4 - 5a^2 - 36 = 0 \]

### Step 7: Let \(x = a^2\)
Let \(x = a^2\), then we have:
\[ x^2 - 5x - 36 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 1\), \(b = -5\), and \(c = -36\):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-36)}}{2 \cdot 1} \]
\[ x = \frac{5 \pm \sqrt{25 + 144}}{2} \]
\[ x = \frac{5 \pm \sqrt{169}}{2} \]
\[ x = \frac{5 \pm 13}{2} \]
This gives us two solutions:
\[ x = \frac{18}{2} = 9 \quad \text{and} \quad x = \frac{-8}{2} = -4 \quad (\text{not valid since } x = a^2 \geq 0) \]
Thus, \(a^2 = 9\) which implies:
\[ a = 3 \quad \text{or} \quad a = -3 \]

### Step 9: Find \(b\)
Using \(b = \frac{6}{a}\):
- If \(a = 3\), then \(b = \frac{6}{3} = 2\).
- If \(a = -3\), then \(b = \frac{6}{-3} = -2\).

### Step 10: Write the square roots
Thus, the two square roots of \(5 + 12i\) are:
\[ 3 + 2i \quad \text{and} \quad -3 - 2i \]

### Final Answer
The square roots of the complex number \(5 + 12i\) are:
\[ 3 + 2i \quad \text{and} \quad -3 - 2i \]