Find the square root fo `(-5+12i)`

Answer

To find the square root of the complex number \(-5 + 12i\), 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 can be expressed as:
\[ \sqrt{-5 + 12i} = x + yi \]
where \(x\) and \(y\) are real numbers.

### Step 2: Square both sides
Squaring both sides gives:
\[ -5 + 12i = (x + yi)^2 \]
Expanding the right-hand side:
\[ (x + yi)^2 = x^2 + 2xyi - y^2 \]
This can be rewritten as:
\[ x^2 - y^2 + 2xyi \]

### Step 3: Equate real and imaginary parts
From the equation \(-5 + 12i = x^2 - y^2 + 2xyi\), we can equate the real and imaginary parts:
1. \(x^2 - y^2 = -5\) (1)
2. \(2xy = 12\) (2)

### Step 4: Solve for \(y\) in terms of \(x\)
From equation (2), we can express \(y\) in terms of \(x\):
\[ y = \frac{12}{2x} = \frac{6}{x} \]

### Step 5: Substitute \(y\) into equation (1)
Substituting \(y\) into equation (1):
\[ x^2 - \left(\frac{6}{x}\right)^2 = -5 \]
This simplifies to:
\[ x^2 - \frac{36}{x^2} = -5 \]
Multiplying through by \(x^2\) to eliminate the fraction:
\[ x^4 + 5x^2 - 36 = 0 \]

### Step 6: Let \(u = x^2\)
Let \(u = x^2\), then we have a quadratic equation:
\[ u^2 + 5u - 36 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ u = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-36)}}{2 \cdot 1} \]
\[ u = \frac{-5 \pm \sqrt{25 + 144}}{2} \]
\[ u = \frac{-5 \pm \sqrt{169}}{2} \]
\[ u = \frac{-5 \pm 13}{2} \]
Calculating the two possible values:
1. \(u = \frac{8}{2} = 4\)
2. \(u = \frac{-18}{2} = -9\) (not valid since \(u = x^2\) must be non-negative)

Thus, \(x^2 = 4\) implies:
\[ x = \pm 2 \]

### Step 8: Find \(y\) using \(x\)
Substituting \(x = 2\) into \(y = \frac{6}{x}\):
\[ y = \frac{6}{2} = 3 \]
Substituting \(x = -2\):
\[ y = \frac{6}{-2} = -3 \]

### Step 9: Write the final results
Thus, the square roots of \(-5 + 12i\) are:
\[ \sqrt{-5 + 12i} = 2 + 3i \quad \text{and} \quad -2 - 3i \]

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