Find the probability of selecting a∈[−5,30] such that x2+2(a+4)x−5a+64>0 for all x∈R.
Solutions · 1
Official solution
Need D<0: 4(a+4)2−4(64−5a)<0⇒a2+13a−48<0⇒(a+16)(a−3)<0⇒−16<a<3.
Within [−5,30]: a∈[−5,2] (integers), giving 8 values, out of 36 total integers.
\[P = \frac{8}{36}=\frac29\]