The probability of selecting integers a ∈ - 5 , 30 such that x 2 + 2 a + 4 x - 5 a + 64 > 0 , for all x ∈ R , is:

Step-by-step solution

We have, x 2 + 2   a + 4   x - 5 a + 64 > 0 If A > 0 and D = B 2 - 4 A C < 0 , then A x 2 + B x + C > 0   ∀ x ∈ R . Hence, D < 0 ⇒     4 a + 4 2 - 4 - 5 a + 64 < 0 ⇒     a 2 + 16 + 8 a + 5 a - 64 < 0 ⇒     a 2 + 13 a - 48 < 0 ⇒     a + 16 a - 3 < 0 ⇒     a ∈ - 16 , 3 ∴ Possible values for a are - 15 ,   - 14 ,   … … . , 2 containing 18 integ