SOLUTION: For an integer n, the inequality x^2 + nx + 15 < -21 - 3x^2 - 85 has no real solutions in x. Find the number of different possible values of n.

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Question 1209299: For an integer n, the inequality
x^2 + nx + 15 < -21 - 3x^2 - 85
has no real solutions in x. Find the number of different possible values of n.
Answer by ikleyn(52848) About Me  (Show Source):
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For an integer n, the inequality
x^2 + nx + 15 < -21 - 3x^2 - 85
has no real solutions in x. Find the number of different possible values of n.
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The given inequality is equivalent to

    4x^2 + nx + 121 < 0.


It has no real solutions if and only if the discriminant 
of the quadratic polynomial in the left side is negative

    d = b^2 - 4ac < 0

    n^2 - 4*4*121 < 0

    n^2 < 16*121

    |n| < sqrt%2816%2A121%29 

    |n| < 4*11

    |n| < 44

    -44 < n < 44.


This inequality has  43 + 1 + 43 = 87 solutions in integer numbers.


ANSWER.  There are 87 different possible integer values of n such that

         the given inequality has no real solutions.

Solved.