Question

[x + 2]² – 9[x] + 2 = 0, where [  ] denotes the greatest integer function​

Answer 1

Answer:

[ X + 2 )*2 -9 [x] + 2 = 0

- X*2 - 4X - 4 +9X - 2

- X*2 + 5X -6

- X*2 - X + 6X + 6 =0

- X ( X + 1 ) + 6 ( X + 1 )

( X + 1 ) ( - X + 6 )

Answer 2

The greatest integer function is defined in such a way that,

$\longrightarrow [x]=a\quad\iff\quad x\in[a,\ a+1)\quad\quad\!\forall a\in\mathbb{Z}$

We're given the equation,

$\longrightarrow [x+2]^2-9[x]+2=0\quad\quad\dots(1)$

Let,

$\longrightarrow [x]=a\quad\iff\quad x\in[a,\ a+1)$

Then we get,

$\longrightarrow [x+2]=a+2\quad\iff\quad x+2\in[a+2,\ a+3)$

Then (1) becomes,

$\longrightarrow (a+2)^2-9a+2=0$

$\longrightarrow a^2+4a+4-9a+2=0$

$\longrightarrow a^2-5a+6=0$

$\longrightarrow a^2-2a-3a+6=0$

$\longrightarrow a(a-2)-3(a-2)=0$

$\longrightarrow (a-2)(a-3)=0$

$\Longrightarrow a\in\{2,\ 3\}$

If $a=2,$

$\longrightarrow x\in[a,\ a+1)$

$\longrightarrow x\in[2,\ 2+1)$

$\longrightarrow x\in[2,\ 3)\quad\quad\dots(2)$

If $a=3,$

$\longrightarrow x\in[a,\ a+1)$

$\longrightarrow x\in[3,\ 3+1)$

$\longrightarrow x\in[3,\ 4)\quad\quad\dots(3)$

On combining (2) and (3), we get,

$\longrightarrow x\in[2,\ 3)\cup[3,\ 4)$

$\longrightarrow\underline{\underline{x\in[2,\ 4)}}$