Suppose `alpha , beta ` are roots of `x^(2) - 2x + 4 = 0 ` If `alpha^(4) + beta^(4) = k` , then |k| is equal to ______

Answer

To solve the problem, we need to find the value of \( |k| \) where \( k = \alpha^4 + \beta^4 \) and \( \alpha, \beta \) are the roots of the equation \( x^2 - 2x + 4 = 0 \).

### Step 1: Find the roots \( \alpha \) and \( \beta \)
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( x^2 - 2x + 4 = 0 \):
- \( a = 1 \)
- \( b = -2 \)
- \( c = 4 \)

Calculating the discriminant:
\[ D = b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = -12 \]
Since the discriminant is negative, the roots are complex:
\[ x = \frac{2 \pm \sqrt{-12}}{2 \cdot 1} = 1 \pm i\sqrt{3} \]
Thus, the roots are:
\[ \alpha = 1 + i\sqrt{3}, \quad \beta = 1 - i\sqrt{3} \]

### Step 2: Calculate \( \alpha^2 + \beta^2 \)
Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):
- \( \alpha + \beta = 2 \)
- \( \alpha\beta = 4 \)

Calculating \( \alpha^2 + \beta^2 \):
\[ \alpha^2 + \beta^2 = (2)^2 - 2 \cdot 4 = 4 - 8 = -4 \]

### Step 3: Calculate \( \alpha^4 + \beta^4 \)
Using the identity \( \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha\beta)^2 \):
\[ \alpha^4 + \beta^4 = (-4)^2 - 2 \cdot (4)^2 \]
Calculating:
\[ \alpha^4 + \beta^4 = 16 - 2 \cdot 16 = 16 - 32 = -16 \]

### Step 4: Find \( |k| \)
Since \( k = \alpha^4 + \beta^4 = -16 \):
\[ |k| = |-16| = 16 \]

Thus, the final answer is:
\[ \boxed{16} \]