Q-prove-that-the-following-equation-has-no-solution-x-x-x-x-1-

Question Number 175653 by mnjuly1970 last updated on 04/Sep/22

Q : p r o v e t h a t t h e f o l l o w i n g

e q u a t i o n h a s n o s o l u t i o n .

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

Answered by mahdipoor last updated on 04/Sep/22

f (x) = \sqrt{x - \sqrt{x}} + \sqrt{x + [x]}

D_{f} = \forall x ⩾ 1

f o r I_{n} = [n, n + 1) n ⩾ 1

f_{n} (x) = \sqrt{x - \sqrt{x}} + \sqrt{x + n}

D_{x} (f_{n} (x)) = \frac{1 - \frac{1}{2 \sqrt{x}}}{2 \sqrt{x - \sqrt{x}}} + \frac{1}{2 \sqrt{x + n}} > 0

f i s i n c r i s e f u n c t i o n i n I_{n} \Rightarrow

R_{f} i n I_{n} = [f_{n} (n), f_{n} (n + 1))

\Rightarrow\Rightarrow

R_{f} i n D_{f} = I_{1} \cap I_{2} \cap \dots =

[\sqrt{2}, \sqrt{3} + \sqrt{2 - \sqrt{2}}) \cap [\sqrt{4} + \sqrt{2 - \sqrt{2}}, \sqrt{5} + \sqrt{3 - \sqrt{3}}) \cap \dots

\Rightarrow \sqrt{2} ⩽ R_{f}

\Rightarrow f (x) \neq 1 f o r \forall x \in D_{f}

Commented by mnjuly1970 last updated on 04/Sep/22

t h a n k s a l o t s i r m a h d i p o o r

Answered by mr W last updated on 04/Sep/22

x ⩾ 0

x ⩾ \sqrt{x} \Rightarrow x ⩾ 1 \Rightarrow ⌊ x ⌋ ⩾ 1

x + ⌊ x ⌋ ⩾ 2

\sqrt{x + ⌊ x ⌋} ⩾ \sqrt{2}

\sqrt{x + ⌊ x ⌋} + \sqrt{x - \sqrt{x}} ⩾ \sqrt{2} > 1

\sqrt{x + ⌊ x ⌋} + \sqrt{x - \sqrt{x}} = 1 c a n n e v e r b e t r u e .

i . e . \sqrt{x + ⌊ x ⌋} + \sqrt{x - \sqrt{x}} = 1 h a s n o s o l u t i o n .

Commented by mnjuly1970 last updated on 04/Sep/22

b r a v o s i r W