Solve-the-following-equation-x-x-x-x-1-x-x-is-floor-of-x-

Question Number 211467 by mnjuly1970 last updated on 10/Sep/24

S o l v e t h e f o l l o w i n g e q u a t i o n

⌊ x ⌋ + \sqrt{x - \sqrt{x}} = ⌊ x + \frac{1}{x} ⌋

⌊ x ⌋ i s f l o o r o f, x,

Answered by Frix last updated on 10/Sep/24

\sqrt{x - \sqrt{x}} \in R \Rightarrow x = 0 \lor x ⩾ 1

⌊ x ⌋ \in N \land ⌊ x + \frac{1}{x} ⌋ \in N \Rightarrow \sqrt{x - \sqrt{x}} \in N

⌊ x + \frac{1}{x} ⌋ - ⌊ x ⌋ = \sqrt{x - \sqrt{x}}

1 . ⌊ x + \frac{1}{x} ⌋ = ⌊ x ⌋

impossible because \sqrt{x - \sqrt{x}} = 0 \Rightarrow

x = 0 \lor x = 1 \Rightarrow ⌊ x + \frac{1}{x} ⌋ \neq ⌊ x ⌋

2 . ⌊ x + \frac{1}{x} ⌋ = ⌊ x ⌋ + 1

\Rightarrow \sqrt{x - \sqrt{x}} = 1 \Rightarrow x = \frac{3 + \sqrt{5}}{2}

Commented by mnjuly1970 last updated on 10/Sep/24