solve-x-x-2-

Question Number 86042 by M±th+et£s last updated on 26/Mar/20

s o l v e : ⌊ \sqrt{x} ⌋ = ⌊ \frac{x}{2} ⌋

Answered by Rio Michael last updated on 26/Mar/20

⌊ \sqrt{x} ⌋ = ⌊ \frac{x}{2} ⌋

\Rightarrow \sqrt{x} = \frac{x}{2}

x = \frac{x^{2}}{4}

x^{2} - 4 x = 0 \Leftrightarrow x = 0 o r x = 4

Commented by M±th+et£s last updated on 26/Mar/20

2^{{l o g}_{2} (x - 2)} = 2^{{l o g}_{2} (2 x - 3)}

x - 2 = 2 x - 3

- x = - 1

x = 1 {l o g}_{2} (1 - 2) = {l o g}_{2} (2 x - 3)

x = \emptyset

Commented by Rio Michael last updated on 26/Mar/20

s a m e p r o c e s s

Commented by M±th+et£s last updated on 26/Mar/20

o k t h a n k y o u s i r

Commented by Rio Michael last updated on 26/Mar/20

how would you solve

\log_{2} (x - 2) = \log_{2} (2 x - 3) ?

Commented by M±th+et£s last updated on 26/Mar/20

s i r w h y ⌊ \sqrt{x} ⌋ = ⌊ \frac{x}{2} ⌋

Commented by M±th+et£s last updated on 26/Mar/20

\Leftrightarrow \sqrt{x} = \frac{x}{2}

Commented by mr W last updated on 26/Mar/20

s o l u t i o n n o t c o r r e c t (c o m p l e t e) s i r!

⌊ A ⌋ = ⌊ B ⌋ {d o e s n}^{'} t m e a n A = B!

e x a m p l e : A = 1.2, B = 1.7

⌊ A ⌋ = ⌊ B ⌋, b u t A \neq B

Commented by mr W last updated on 26/Mar/20

⌊ \sqrt{3} ⌋ = ⌊ \frac{3}{2} ⌋ \Rightarrow x = 3 i s a l s o s o l u t i o n, e t c .

Commented by M±th+et£s last updated on 27/Mar/20

t h a n k y o u s i r b u t i f w e s a y

⌊ A ⌋ = ⌊ B ⌋ m e a n s A = B t h e s o l u t i o n w i l l b e

w r o n g o r n o t c o m p l e t e a n d w h y

Commented by mr W last updated on 27/Mar/20

i f ⌊ A ⌋ = ⌊ B ⌋ i t m e a n s

A = n + d w i t h 0 ⩽ d < 1

B = n + δ w i t h 0 ⩽ δ < 1

i f y o u s a y A = B, i t i s w r o n g i n t h e

s e n s e o f m a t h e m a t i c s .

i f A = B \Rightarrow ⌊ A ⌋ = ⌊ B ⌋

b u t i f ⌊ A ⌋ = ⌊ B ⌋ ⇏ A = B

Commented by M±th+et£s last updated on 27/Mar/20

t h a n k y o u s i r

Answered by Rio Michael last updated on 26/Mar/20

⌊ \sqrt{x} ⌋ = ⌊ \frac{x}{2} ⌋

\Rightarrow \sqrt{x} = \frac{x}{2}

x = \frac{x^{2}}{4}

x^{2} - 4 x = 0 \Leftrightarrow x = 0 o r x = 4

Answered by mr W last updated on 27/Mar/20

x ⩾ 0

l e t ⌊ \sqrt{x} ⌋ = ⌊ \frac{x}{2} ⌋ = n \in N

n = 0 :

\Rightarrow 0 ⩽ x < 1 \Rightarrow s o l u t i o n

\Rightarrow 0 ⩽ x < 2

n = 1 :

\Rightarrow 1 ⩽ x < 4

\Rightarrow 2 ⩽ x < 4 \Rightarrow s o l u t i o n

n = 2 :

\Rightarrow 4 ⩽ x < 9

\Rightarrow 4 ⩽ x < 6 \Rightarrow s o l u t i o n

n = 3 :

\Rightarrow 9 ⩽ x < 16

\Rightarrow 6 ⩽ x < 8

\Rightarrow n o s o l u t i o n u p o n n ⩾ 3

s o l u t i o n i s :

0 ⩽ x < 1 \lor 2 ⩽ x < 6

Commented by M±th+et£s last updated on 27/Mar/20

g o d b l e s s y o u s i r