The number of integral values of k for which the inequality
`x^(2)-2(4k-1)x+15k^(2)-2k-7 ge0` hold for `x in R` are _____________

Options

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

Answer

To solve the inequality \( x^2 - 2(4k-1)x + (15k^2 - 2k - 7) \geq 0 \) for integral values of \( k \), we will follow these steps:

### Step 1: Identify the coefficients of the quadratic
The given quadratic can be expressed in the standard form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -2(4k - 1) = -8k + 2 \)
- \( c = 15k^2 - 2k - 7 \)

### Step 2: Determine the condition for the quadratic to be non-negative
For the quadratic to be non-negative for all \( x \in \mathbb{R} \), the discriminant \( D \) must be less than or equal to zero:
\[ D = b^2 - 4ac \leq 0 \]

### Step 3: Calculate the discriminant
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-8k + 2)^2 - 4 \cdot 1 \cdot (15k^2 - 2k - 7) \]
Calculating \( D \):
\[ D = (64k^2 - 32k + 4) - (60k^2 - 8k - 28) \]
\[ D = 64k^2 - 32k + 4 - 60k^2 + 8k + 28 \]
\[ D = (64k^2 - 60k^2) + (-32k + 8k) + (4 + 28) \]
\[ D = 4k^2 - 24k + 32 \]

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

### Step 5: Factor the quadratic
Factoring gives:
\[ (k - 2)(k - 4) \leq 0 \]

### Step 6: Determine the intervals
The roots of the equation \( (k - 2)(k - 4) = 0 \) are \( k = 2 \) and \( k = 4 \). The quadratic opens upwards (since the coefficient of \( k^2 \) is positive), so the expression is non-positive between the roots:
\[ 2 \leq k \leq 4 \]

### Step 7: Identify the integral values of \( k \)
The integral values of \( k \) in the interval \( [2, 4] \) are:
- \( k = 2 \)
- \( k = 3 \)
- \( k = 4 \)

### Conclusion
Thus, the number of integral values of \( k \) for which the inequality holds is **3**.

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