The-possible-value-of-x-that-satisfy-x-2-x-2-2x-x-2-2x-1-2017-x-R-and-x-2-2x-

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 ⌉ = \dots

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 \dots

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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