Solve-x-3-x-3-

Question Number 15501 by rish@bh last updated on 11/Jun/17

Solve

x^{3} - ⌊ x ⌋ = 3

Answered by mrW1 last updated on 11/Jun/17

x = n + f

⌊ x ⌋ = n

n + 3 = {(n + f)}^{3}

n^{3} ⩽ {(n + f)}^{3} < {(n + 1)}^{3} = n^{3} + {3 n}^{2} + 3 n + 1

n^{3} ⩽ n + 3 < n^{3} + {3 n}^{2} + 3 n + 1

n^{3} - n ⩽ 3

(n - 1) n (n + 1) ⩽ 3

\Rightarrow n < 2

n^{3} + {3 n}^{2} + 2 n > 2

n (n + 1) (n + 2) > 2

\Rightarrow n ⩾ 1

that means the solution

1 < x < 2

⌊ x ⌋ = 1

x^{3} - 1 = 3

\Rightarrow x =^{3} \sqrt{4}

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