1. Discriminant (D) must be greater than zero
For real and distinct roots: D > 0
8^2 - 4(k - 2)(k + 4) > 0
64 - 4(k^2 + 2k - 8) > 0
Divide by 4:
16 - (k^2 + 2k - 8) > 0
-k^2 - 2k + 24 > 0 \implies k^2 + 2k - 24 < 0
Factors: (k + 6)(k - 4) < 0
So, -6 < k < 4 (Condition 1)
2. Sum of roots must be negative
Let roots be α and β. If both are negative, \alpha + \beta < 0.
\frac{-b}{a} < 0 \implies \frac{-8}{k - 2} < 0
For this fraction to be negative, the denominator (k−2) must be positive:
k - 2 > 0 \implies \mathbf{k > 2}
(Condition 2)
3. Product of roots must be positive
If both roots are negative, their product \alpha\beta > 0.
\frac{c}{a} > 0 \implies \frac{k + 4}{k - 2} > 0
From Condition 2, we already know k - 2 > 0. Therefore, for the fraction to be positive:
k + 4 > 0 \implies \mathbf{k > -4}
(Condition 3)
4. Intersection of all conditions
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Condition 1: k∈(−6,4)
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Condition 2: k > 2
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Condition 3: k > -4
The intersection of these ranges is:
Looking at the given options:
(a) 0 (False)
(b) 2 (False, as k must be strictly greater than 2)
(c) 3 (True, as 3 lies between 2 and 4)
(d) −4 (False)
Correct Option: (c)
