If the inequality x 2 + a x + a 2 + 6 a < 0 is satisfied for all x ∈ 1,2 , then the sum of all the integral values of a must be equal to

Step-by-step solution

f x = x 2 + a x + a 2 + 6 a f 1 < 0 ,   f 2 < 0 and D > 0 f 1 < 0 ⇒ a 2 + 7 a + 1 < 0 ⇒ a ∈ - 7 - 3 5 2 , - 7 + 3 5 2 f 2 < 0 ⇒ a 2 + 8 a + 4 < 0 ⇒ a ∈ - 4 - 2 3 , - 4 + 2 3 and D > 0 ⇒ a 2 - 4 × 1 × a 2 + 6 a > 0 ⇒ - 3 a 2 - 24 a > 0 ⇒ a 2 + 8 a < 0 ⇒ a ∈ - 8,0 Taking intersection, we get, a ∈ - 7 - 3 5 2 , - 4 + 2 3 Integral values of a = - 6 , - 5 , - 4 , - 3 , - 2 , - 1 Hence, the s