If the product of the roots of the equation `x^(2)-5kx+2e^("ln|k|")-1=0` is 49, then the sum of the squares of the roots of the equation is :

Options

A. 1524
B. 1152
C. 1244
D. 1654

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

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

### Step 2: Use the properties of roots of a quadratic equation
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots \( \alpha + \beta \) and the product of the roots \( \alpha \beta \) can be expressed as:
- Sum of the roots: \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots: \( \alpha \beta = \frac{c}{a} \)

In our case:
- \( a = 1 \)
- \( b = -5k \)
- \( c = 2e^{\ln |k|} - 1 \)

Thus, we have:
- \( \alpha + \beta = -\frac{-5k}{1} = 5k \)
- \( \alpha \beta = \frac{2e^{\ln |k|} - 1}{1} = 2e^{\ln |k|} - 1 \)

### Step 3: Simplify the product of the roots
We know from the problem statement that the product of the roots \( \alpha \beta = 49 \). Therefore, we can set up the equation:
\[ 2e^{\ln |k|} - 1 = 49 \]

### Step 4: Solve for \( k \)
Rearranging the equation gives:
\[ 2e^{\ln |k|} = 50 \]
\[ e^{\ln |k|} = 25 \]
Using the property \( e^{\ln x} = x \), we have:
\[ |k| = 25 \]
Thus, \( k^2 = 625 \).

### Step 5: Find the sum of the squares of the roots
The sum of the squares of the roots can be calculated using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

Substituting the values we found:
- \( \alpha + \beta = 5k \)
- \( \alpha \beta = 49 \)

Now substituting these into the identity:
\[ \alpha^2 + \beta^2 = (5k)^2 - 2 \cdot 49 \]
\[ = 25k^2 - 98 \]

### Step 6: Substitute \( k^2 \) into the equation
We already found \( k^2 = 625 \). Thus:
\[ \alpha^2 + \beta^2 = 25 \cdot 625 - 98 \]
\[ = 15625 - 98 \]
\[ = 15527 \]

### Final Answer
The sum of the squares of the roots is:
\[ \boxed{15527} \]