Solve the equation `x^(2)+ 2ix+15=0`.

Explanation

To solve the equation \( x^2 + 2ix + 15 = 0 \), we can use the quadratic formula or factorization. Let's proceed step by step using the quadratic formula.

### Step 1: Identify the coefficients
The given equation is in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).
Here,
- \( a = 1 \)
- \( b = 2i \)
- \( c = 15 \)

### Step 2: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Calculate the discriminant
First, we need to calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (2i)^2 = 4i^2 = 4(-1) = -4 \]
\[ 4ac = 4 \cdot 1 \cdot 15 = 60 \]
Now, substituting these values into the discriminant:
\[ b^2 - 4ac = -4 - 60 = -64 \]

### Step 4: Substitute into the quadratic formula
Now substitute \( b \), \( a \), and the discriminant into the quadratic formula:
\[ x = \frac{-2i \pm \sqrt{-64}}{2 \cdot 1} \]

### Step 5: Simplify the square root
We know that \( \sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i \). Thus, we have:
\[ x = \frac{-2i \pm 8i}{2} \]

### Step 6: Split into two cases
We can split this into two cases:
1. \( x = \frac{-2i + 8i}{2} = \frac{6i}{2} = 3i \)
2. \( x = \frac{-2i - 8i}{2} = \frac{-10i}{2} = -5i \)

### Step 7: Write the final solutions
The solutions to the equation \( x^2 + 2ix + 15 = 0 \) are:
\[ x = 3i \quad \text{and} \quad x = -5i \]