Integral value of k for which `x^2-2(3k-1)x+8k^2-7 gt 0`

Answer

To find the integral value of \( k \) for which the inequality \( x^2 - 2(3k-1)x + (8k^2 - 7) > 0 \) holds for all \( x \), we need to ensure two conditions are satisfied:

1. The coefficient of \( x^2 \) must be greater than zero.
2. The discriminant of the quadratic must be less than zero.

### Step 1: Check the coefficient of \( x^2 \)

The coefficient of \( x^2 \) is \( 1 \), which is greater than zero. This condition is satisfied.

**Hint:** Always check the leading coefficient of the quadratic to ensure it is positive.

### Step 2: Calculate the discriminant

The discriminant \( D \) of the quadratic equation \( ax^2 + bx + c \) is given by the formula:
\[ D = b^2 - 4ac \]
In our case:
- \( a = 1 \)
- \( b = -2(3k - 1) = -6k + 2 \)
- \( c = 8k^2 - 7 \)

Substituting these values into the discriminant formula:
\[ D = (-6k + 2)^2 - 4(1)(8k^2 - 7) \]

### Step 3: Expand the discriminant

Calculating \( D \):
\[ D = (36k^2 - 24k + 4) - (32k^2 - 28) \]
\[ D = 36k^2 - 24k + 4 - 32k^2 + 28 \]
\[ D = (36k^2 - 32k^2) + (-24k) + (4 + 28) \]
\[ D = 4k^2 - 24k + 32 \]

### Step 4: Set the discriminant less than zero

To ensure the quadratic is always positive, we need:
\[ 4k^2 - 24k + 32 < 0 \]
Dividing the entire inequality by 4:
\[ k^2 - 6k + 8 < 0 \]

### Step 5: Factor the quadratic

Factoring the quadratic:
\[ k^2 - 6k + 8 = (k - 4)(k - 2) \]
Thus, we need:
\[ (k - 4)(k - 2) < 0 \]

### Step 6: Determine the intervals

The roots of the equation are \( k = 2 \) and \( k = 4 \). We can test the intervals:
- For \( k < 2 \): both factors are negative, product is positive.
- For \( 2 < k < 4 \): one factor is negative and the other is positive, product is negative.
- For \( k > 4 \): both factors are positive, product is positive.

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

### Step 7: Find integral values of \( k \)

The integral value of \( k \) that lies in the interval \( (2, 4) \) is:
\[ k = 3 \]

### Final Answer

The integral value of \( k \) for which \( x^2 - 2(3k - 1)x + (8k^2 - 7) > 0 \) for all \( x \) is:
\[ \boxed{3} \]