Question

Let $x$ be a real number $\left[x\right]$ denotes the greatest integer function, and $\left\{x\right\}$ denotes the fractional part and $\left(x\right)$ denotes the least integer function,then solve the following.${\left[x\right]}^2+2\left[x\right]=3x,0\le x\le 2$

• A. $\{0,1\}$
• B. $\{-1,1\}$
• C. $\{0,-1\}$
• D. $\{2,1\}$

Answer 1

The correct option is A $\{0,1\}$

Given, ${\left[x\right]}^2+2\left[x\right]=3x$

Case 1. when $0\le x<1\Rightarrow \left[x\right]=0$
$\therefore 3x=0+0\Rightarrow x=0$

Case 2. when $1\le x<2\Rightarrow \left[x\right]=1$
$\therefore 3x=1+2\Rightarrow x=1$

Case 3. when $x=2\Rightarrow \left[x\right]=2$
$\therefore 3x=2^2+2.2=8\Rightarrow x=8/3\ne 2$

Hence solution is $\{0,1\}$