The sum of [x] for all real numbers $x$ satisfying the equation 16 + 15x + 15x2 = [x]3 is:

Step 1: [x]3 = 15x2 + 15x + 16

Step 2: [x] in (x – 1, x]

Step 3: [x]3 in ((x – 1)3, x3]

Step 4: => 15x2 + 15x + 16 in ((x – 1)3, x3]

Step 5: x3  15x2 + 15x + 16

Step 6: => x3 – 15x2 – 15x – 16  0

Step 7: (x – 16)(x2 + $x$ + 1)  0

Step 8: => $x$  16

Step 9: Also, 15x2 + 15x + 16 > x3 – 3x2 + 3x – 1

Step 10: => x3 – 18x2 – 12x – 17 < 0

Step 11: Let f(x) = x3 – 18x2 – 12x – 17

Step 12: f(x) = 3x2 – 36x – 12

Step 13: => f(x) has only one real root

Step 14: => f(18) = –233

Step 15: f(19) = 116

Step 16: => f() = 0

Step 17: =>  in (18, 19)

Step 18: For this equation

Step 19: x in [16, 19)

Step 20: => [x] = 16, 17, 18

Step 21: (i) [x] = 16

Step 22: => 15x2 + 15x + 16 = 163

Step 23: (x – 16)(x + 17) = 0

Step 24: => $x$ = 16 satisfies

Step 25: (ii) [x] = 17

Step 26: => 15x2 + 15x + 16 = 173

Step 27: −15  152 − 4.16.15

Step 28: => [x] = 17 satisfies

Step 29: (iii) [x] = 18

Step 30: => $x$ > 19 and $x$ < –20

Step 31: => No received value of x

Step 32: => Sum of [x] = 16 + 17 = 33.