25. The integer k for which the inequality `x²-2(4k-1)x +15k²- 2k-7 > 0 is valid for any real x is
(a) 2
(b) 3
(c) 4
(d) infinite

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the coefficients:**
The given quadratic can be expressed in 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 \( D \) of a quadratic equation \( ax^2 + bx + c \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-8k + 2)^2 - 4 \cdot 1 \cdot (15k^2 - 2k - 7) \]

3. **Expand the discriminant:**
First, calculate \( (-8k + 2)^2 \):
\[ D = (64k^2 - 32k + 4) - 4(15k^2 - 2k - 7) \]
Now expand the second part:
\[ D = 64k^2 - 32k + 4 - (60k^2 - 8k - 28) \]
Combine like terms:
\[ D = 64k^2 - 60k^2 - 32k + 8k + 4 + 28 \]
This simplifies to:
\[ D = 4k^2 - 24k + 32 \]

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

5. **Factor the quadratic:**
Factor the quadratic:
\[ (k - 2)(k - 4) \leq 0 \]

6. **Determine the intervals:**
The critical points are \( k = 2 \) and \( k = 4 \). We can test intervals around these points:
- For \( k < 2 \): Choose \( k = 1 \) → \( (1-2)(1-4) > 0 \) (not valid)
- For \( 2 \leq k \leq 4 \): Choose \( k = 3 \) → \( (3-2)(3-4) < 0 \) (valid)
- For \( k > 4 \): Choose \( k = 5 \) → \( (5-2)(5-4) > 0 \) (not valid)

7. **Conclusion:**
The inequality \( (k - 2)(k - 4) \leq 0 \) holds for \( k \) in the interval:
\[ k \in [2, 4] \]
Since \( k \) must be an integer, the possible values of \( k \) are \( 2, 3, \) and \( 4 \).

### Final Answer:
The integer \( k \) for which the inequality is valid for any real \( x \) is \( 2, 3, \) or \( 4 \).