Find the number of integers n that satisfy \(7 \sqrt{-n^2 + 22n - 21} \le n + 39.\)
+23246 Which integers satisfy: 7 · sqrt( -n2 + 22n - 21 ) <= n + 39
square both sides: 49 · ( -n2 + 22n - 21 ) < = (n + 39)2
-49n2 + 1078n - 1029 <= n2 + 78n + 1521
0 <= 50n2 - 1000n + 2550
0 <= n2 - 20 n + 51
0 <= (n - 17)(n - 3)
Either n <= 3 or n >= 17
Also: -n2 + 22n - 21 >= 0
n2 - 22n + 21 <= 0
So: n >= 1 and n <= 21
Combining these: n = 1, 2, 3, 17, 18, 19, 20, 21
