The roots of `x^(2)-7ix-12=0` are . . . And . . . .

Answer

To find the roots of the quadratic equation \( x^2 - 7ix - 12 = 0 \), we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify coefficients
In the equation \( x^2 - 7ix - 12 = 0 \), we identify the coefficients:
- \( a = 1 \)
- \( b = -7i \)
- \( c = -12 \)

### Step 2: Substitute coefficients into the quadratic formula
Now we substitute these values into the quadratic formula:

\[ x = \frac{-(-7i) \pm \sqrt{(-7i)^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \]

### Step 3: Simplify the expression
This simplifies to:

\[ x = \frac{7i \pm \sqrt{(-7i)^2 + 48}}{2} \]

### Step 4: Calculate \( b^2 \)
Now, we calculate \( (-7i)^2 \):

\[ (-7i)^2 = 49i^2 = 49(-1) = -49 \]

### Step 5: Substitute back into the equation
Substituting this back into our expression gives:

\[ x = \frac{7i \pm \sqrt{-49 + 48}}{2} \]

### Step 6: Simplify under the square root
Now simplify under the square root:

\[ -49 + 48 = -1 \]

Thus, we have:

\[ x = \frac{7i \pm \sqrt{-1}}{2} \]

### Step 7: Substitute \( \sqrt{-1} \) with \( i \)
Since \( \sqrt{-1} = i \), we can write:

\[ x = \frac{7i \pm i}{2} \]

### Step 8: Separate the two cases
This gives us two cases:

1. \( x = \frac{7i + i}{2} = \frac{8i}{2} = 4i \)
2. \( x = \frac{7i - i}{2} = \frac{6i}{2} = 3i \)

### Final Roots
Thus, the roots of the equation \( x^2 - 7ix - 12 = 0 \) are:

\[ x = 4i \quad \text{and} \quad x = 3i \]