Key Concepts and Formulas

• For a quadratic $ax^2 + bx + c$ to be strictly positive for all real $x$, we need $a > 0$ and the discriminant $D = b^2 - 4ac < 0$.
• Solving quadratic inequalities involves finding the roots of the corresponding quadratic equation and then determining the intervals where the inequality holds.
• Factoring quadratic expressions to find roots.

Step-by-Step Solution

Step 1: Understanding the Problem and Identifying Coefficients

We are given the quadratic inequality $x^2 - 2(3k - 1)x + 8k^2 - 7 > 0$ and need to find the integer value of $k$ for which this inequality holds for all real $x$. We first identify the coefficients:

• $a = 1$
• $b = -2(3k - 1)$
• $c = 8k^2 - 7$

Step 2: Checking the Leading Coefficient Condition

For the quadratic to be always positive, the leading coefficient $a$ must be positive. Since $a = 1 > 0$, this condition is satisfied.

Step 3: Applying the Discriminant Condition

The discriminant, $D = b^2 - 4ac$, must be less than 0 for the inequality to hold for all real $x$. Substituting the coefficients: $D = (-2(3k - 1))^2 - 4(1)(8k^2 - 7) < 0$

Step 4: Simplifying the Discriminant

We simplify the expression for $D$: $D = 4(3k - 1)^2 - 4(8k^2 - 7) < 0$ $D = 4(9k^2 - 6k + 1) - 4(8k^2 - 7) < 0$ $D = 36k^2 - 24k + 4 - 32k^2 + 28 < 0$ $D = 4k^2 - 24k + 32 < 0$

Step 5: Simplifying the Inequality

Divide the inequality by 4: $k^2 - 6k + 8 < 0$

Step 6: Factoring the Quadratic Expression

Factor the quadratic expression: $(k - 2)(k - 4) < 0$

Step 7: Finding the Roots

The roots of the corresponding quadratic equation $(k - 2)(k - 4) = 0$ are $k = 2$ and $k = 4$.

Step 8: Determining the Interval for k

Since the parabola $k^2 - 6k + 8$ opens upwards, the expression is negative between the roots. Thus, we need: $2 < k < 4$

Step 9: Finding the Integer Value of k

The only integer value of $k$ that satisfies the inequality $2 < k < 4$ is $k = 3$.

Common Mistakes & Tips

• Remember to check the sign of the leading coefficient 'a'. If 'a' is not positive, the condition for the quadratic to be greater than zero for all x will not hold.
• Ensure the discriminant is strictly less than zero ($D < 0$) and not less than or equal to zero ($D \le 0$). If $D=0$, the quadratic will touch the x-axis and thus not be strictly greater than zero for all x.
• Double-check factorization and root-finding to avoid errors in determining the interval for $k$.

Summary

We found the integer value of $k$ for which the quadratic inequality holds true for all real numbers $x$. By applying the conditions $a > 0$ and $D < 0$, we arrived at the inequality $2 < k < 4$. The only integer satisfying this condition is $k = 3$.

Final Answer

The final answer is \boxed{3}, which corresponds to option (C).