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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 \(n\) .

 Dec 10, 2023
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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.

 Dec 10, 2023

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