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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| <
|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.