1. First Condition (a > 0):
In our equation, a=k−2.
2. Second Condition (D < 0):
The discriminant is D=b2−4ac. Here, a=(k−2), b=8, and c=(k+4).
8^2 - 4(k - 2)(k + 4) < 0
64 - 4(k^2 + 4k - 2k - 8) < 0
64 - 4(k^2 + 2k - 8) < 0
Divide the entire inequality by 4 to simplify:
16 - (k^2 + 2k - 8) < 0
Multiply by −1 (and reverse the inequality sign):
3. Solve the Quadratic Inequality:
Factorize the expression:
The critical values are k=−6 and k=4. For the expression to be greater than zero:
k < -6 \quad \text{or} \quad k > 4
4. Combine All Conditions:
We have:
The intersection of these conditions is k > 4.
5. Find the Least Integral Value:
Since k must be an integer and k is strictly greater than 4, the smallest integer value is:
The correct option is (a) 5.
