Q. 18

# Mark the Correct alternative in the following:

The set of all vales of m for which both the roots of the equation are real and negative, is

A. (−,−3] [5, ∞)

B. [−3, 5]

C. (−4, −3]

D. (−3, −1]

Answer :

For roots to be real its D ≥ 0

(m + 1)^{2} – 4(m + 4) ≥ 0

m^{2} – 2m – 15 ≥ 0

(m – 1)^{2} – 16 ≥ 0

(m – 1)^{2} ≥ 16

m – 1 ≤ -4 or m – 1 ≥ 4

m ≤ -3 or m ≥ 5

For both roots to be negative product of roots should be

positive and sum of roots should be negative.

Product of roots = m + 4 > 0 ⇒ m > -4

Sum of roots = m + 1 < 0 ⇒ m < -1

After taking intersection of D ≥ 0, Product of roots > 0 and

sum of roots < 0. We can say that the final answer is

m ∈ (-4, -3]

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