The possible value of x that satisfy : x^2 + ⌊x^2 − 2x⌋ + ⌈x^2 + 2x + 1⌉ = 2017 , x ∈ R^+ and ⌊x^2 ⌋ − ⌈2x⌉ = ...


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Question Number 56719 by naka3546 last updated on 22/Mar/19

$T h e p o s s i b l e v a l u e o f x t h a t s a t i s f y :$ $x^{2} + ⌊ x^{2} - 2 x ⌋ + ⌈ x^{2} + 2 x + 1 ⌉ = 2017, x \in R^{+}$ $a n d ⌊ x^{2} ⌋ - ⌈ 2 x ⌉ = . . .$
Commented by tanmay.chaudhury50@gmail.com last updated on 22/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 22/Mar/19

$x = 25.923$ $⌊ {(25.923)}^{2} ⌋ - ⌈ 2 \times 25.923 ⌉$ $= ⌊ 672.001929 ⌋ - ⌈ 51.846 ⌉$ $= 672 - 52$ $= 620$
Commented by naka3546 last updated on 22/Mar/19

$S i r, t h e a n s w e r x n o t s u i t a b l e f o r t h e e q u a t i o n a b o v e .$ $P l e a s e, r e c h e c k t h e a n s w e r .$
Commented by tanmay.chaudhury50@gmail.com last updated on 22/Mar/19

$o k l e t m e c h e c k . . .$
	Answered by mr W last updated on 23/Mar/19

	$x^{2} + ⌊ x^{2} - 2 x ⌋ + ⌈ x^{2} + 2 x + 1 ⌉ = 2017$ $x^{2} + i n t e g e r + i n t e g e r = i n t e g e r$ $\Rightarrow x^{2} = i n t e g e r, s a y x^{2} = n$ $n + n + ⌊ - 2 \sqrt{n} ⌋ + n + ⌈ 2 \sqrt{n} ⌉ + 1 = 2017$ $s i n c e ⌊ - x ⌋ = - ⌈ x ⌉$ $\Rightarrow ⌊ - 2 \sqrt{n} ⌋ + ⌈ 2 \sqrt{n} ⌉ = 0$ $\Rightarrow 3 n = 2016$ $\Rightarrow n = 672$ $\Rightarrow x = \sqrt{n} = \sqrt{672} = 4 \sqrt{42} \approx 25.923$ $⌊ x^{2} ⌋ - ⌈ 2 x ⌉ = 672 - 52 = 620$
	Commented by naka3546 last updated on 23/Mar/19

	$b u t, i f x i s s u b s t i t u t e d t o e q u a t i o n a b o v e .$ $I s x r i g h t ?$ $x^{2} + ⌊ x^{2} - 2 x ⌋ + ⌈ x^{2} + 2 x + 1 ⌉ = 2015$ $h o w i s i t h a p p e n ?$
	Commented by mr W last updated on 23/Mar/19

	$h o w d i d y o u g e t 2015 ? t h i s i s h o w i t i s :$ $x^{2} = 672$ $x = \sqrt{672} \approx 25.92$ $2 x \approx 51.8$ $⌊ x^{2} - 2 x ⌋ = ⌊ 672 - 51.8 ⌋ = ⌊ 620.2 ⌋ = 620$ $⌈ x^{2} + 2 x + 1 ⌉ = ⌈ 672 + 51.8 + 1 ⌉ = ⌈ 724.8 ⌉ = 725$ $x^{2} + ⌊ x^{2} - 2 x ⌋ + ⌈ x^{2} + 2 x + 1 ⌉ = 672 + 620 + 725 = 2017$ $s o x = \sqrt{672} i s c o r r e c t .$
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