If the product of the roots of the equation `x^(2) - 5 kx + 2e^(4"Ink") - 1 = 0` is 31, then sum of the root is

Options

A. `-10`
B. 5
C. `-8`
D. 10`

Answer

To solve the problem step by step, we will analyze the given quadratic equation and use the properties of roots.

### Step 1: Identify the quadratic equation
The given quadratic equation is:
\[ x^2 - 5kx + (2e^{4 \ln k} - 1) = 0 \]

### Step 2: Simplify the term involving \( e^{4 \ln k} \)
Using the property \( e^{\ln a} = a \), we can simplify \( e^{4 \ln k} \) as follows:
\[ e^{4 \ln k} = k^4 \]
Thus, we can rewrite the equation as:
\[ x^2 - 5kx + (2k^4 - 1) = 0 \]

### Step 3: Use the product of the roots
The product of the roots \( \alpha \) and \( \beta \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \alpha \beta = \frac{c}{a} \]
In our case:
- \( a = 1 \)
- \( b = -5k \)
- \( c = 2k^4 - 1 \)

Thus, the product of the roots is:
\[ \alpha \beta = \frac{2k^4 - 1}{1} = 2k^4 - 1 \]

### Step 4: Set the product of the roots equal to 31
According to the problem, the product of the roots is given to be 31:
\[ 2k^4 - 1 = 31 \]

### Step 5: Solve for \( k \)
Now, we can solve for \( k \):
\[ 2k^4 - 1 = 31 \\ 2k^4 = 32 \\ k^4 = 16 \\ k = 2 \quad (\text{since } k \text{ must be positive}) \]

### Step 6: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-5k}{1} = 5k \]
Substituting \( k = 2 \):
\[ \alpha + \beta = 5 \times 2 = 10 \]

### Final Answer
The sum of the roots is:
\[ \boxed{10} \]