The integer k for which the inequality `x^(2) - 2(4k-1)x + 15k^(2) - 2k - 7 gt 0` is valid for any x is :

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the inequality \( x^2 - 2(4k-1)x + (15k^2 - 2k - 7) > 0 \) for any \( x \), we need to ensure that the quadratic expression is always positive. This occurs when the discriminant of the quadratic is less than zero.

### Step-by-step Solution:

1. **Identify the coefficients**:
The given quadratic can be compared to the standard form \( ax^2 + bx + c \):
- \( a = 1 \)
- \( b = -2(4k - 1) = -8k + 2 \)
- \( c = 15k^2 - 2k - 7 \)

2. **Calculate the discriminant**:
The discriminant \( D \) is given by the formula \( D = b^2 - 4ac \). We need to set up the inequality:
\[ D < 0 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-8k + 2)^2 - 4(1)(15k^2 - 2k - 7) \]

3. **Expand the discriminant**:
\[ D = (64k^2 - 32k + 4) - (60k^2 - 8k - 28) \]
Simplifying this:
\[ D = 64k^2 - 32k + 4 - 60k^2 + 8k + 28 \]
\[ D = (64k^2 - 60k^2) + (-32k + 8k) + (4 + 28) \]
\[ D = 4k^2 - 24k + 32 \]

4. **Set the discriminant less than zero**:
We need:
\[ 4k^2 - 24k + 32 < 0 \]
Dividing the entire inequality by 4:
\[ k^2 - 6k + 8 < 0 \]

5. **Factor the quadratic**:
The quadratic can be factored as:
\[ (k - 2)(k - 4) < 0 \]

6. **Determine the intervals**:
The critical points are \( k = 2 \) and \( k = 4 \). We analyze the sign of the product \( (k - 2)(k - 4) \) in the intervals:
- \( k < 2 \): Positive
- \( 2 < k < 4 \): Negative
- \( k > 4 \): Positive

Thus, the inequality \( (k - 2)(k - 4) < 0 \) holds for:
\[ 2 < k < 4 \]

7. **Identify integer values**:
The integers in the interval \( (2, 4) \) are:
\[ k = 3 \]

### Final Answer:
The integer \( k \) for which the inequality is valid for any \( x \) is:
\[ \boxed{3} \]