For an integer \(n\), the inequality \([x^2 + nx + 15 < -21]\)

has no real solutions in \(x\) . Find the number of different possible values of \(n\) .

Ookbroo Dec 10, 2023

#1**0 **

In order for the quadratic x2+nx+15 < -21 to have no real solutions, its discriminant must be negative. The discriminant is n^2−4⋅1⋅15=n^2−60.

For the quadratic to have no real solutions, we need n^2−60<0. We solve this inequality to get n^2<60, or −sqrt(60)

Since n is an integer, the only possible values of n are −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, and 7. Therefore, there are 15 possible values of n.

BuiIderBoi Dec 10, 2023