solve-x-1-x-2-x-1-x-2-0-

Question Number 160558 by mnjuly1970 last updated on 01/Dec/21

s o l v e

⌊ x - \sqrt{1 - x^{2}} ⌋ + ⌊ x + \sqrt{1 - x^{2}} ⌋ = 0

Answered by mr W last updated on 02/Dec/21

c a s e 1 : 0 + 0 = 0

x - \sqrt{1 - x^{2}} ⩾ 0

x + \sqrt{1 - x^{2}} < 1

x ⩾ \sqrt{1 - x^{2}} \Rightarrow x ⩾ \frac{1}{\sqrt{2}}

\sqrt{1 - x^{2}} < 1 - x \Rightarrow x < 0

\Rightarrow c o n t r a d i c t i o n \times

c a s e 2 : - 1 + 1 = 0

x - \sqrt{1 - x^{2}} < 0

x + \sqrt{1 - x^{2}} ⩾ 1

x < \sqrt{1 - x^{2}} \Rightarrow x < \frac{1}{\sqrt{2}}

\sqrt{1 - x^{2}} ⩾ 1 - x \Rightarrow x ⩾ 0

\Rightarrow o k ✓

s o l u t i o n i s : 0 ⩽ x < \frac{1}{\sqrt{2}}

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