+1839 Find the number of integers $n$ that satisfy $n^2 < 64 - 20n - n^2.$
+129895 Simplify as
2n^2 + 20n - 64 < 0
n^2 + 10n - 32 < 0 set up as an equality
n^2 + 10n = 32 complete the square on n
n^2 + 10n + 25 = 32 + 25
( n + 5)^2 = 57 take both roots
n + 5 = sqrt (57) and n + 5 = -sqrt (57)
n = sqrt (57) - 5 ≈ 2.5 n = -5 -sqrt (57) ≈ -12.5
These are roots of the quadratic which opens upward.....every integer between them will cause the given inequality to be < 0
The number of intgers = 2 - -12 + 1 = 2 + 12 + 1 = 15
