+1791 For an integer $n$, the inequality
\[x^2 + nx + 15 < -21 - x^2 - 150 \]
has no real solutions in $x$. Find the number of different possible values of $n$.
+1791 Combining like terms, we have the inequality 2x2+nx−135<0. We can use the quadratic formula to solve for the roots of 2x2+nx−135=0.
The quadratic formula gives us:
x=2⋅2−n±n2−4⋅2⋅−135
x=4−n±n2+1080
Since we are given that the inequality has no real solutions, the discriminant n2+1080 must be negative. This gives us the inequality n2<−1080.
Since n is an integer, the only possible values of n are −34, −33, −32, ..., 32, 33, and 34. Therefore, there are 67 possible values of n.
