Question

The total number of solutions of $[x{]}^2=x+2(x\},$ where [.] and {.} denote the greatest integer function and the fractional part function, respectively, is equal to

• A. 2
• B. 3
• C. 4
• D. 6

Answer 1

The correct option is C 4

${\left(x\right]}^2=x+2\left(x\right\}\Rightarrow {\left(x\right]}^2=\left(x\right]+\left(x\right\}+2\left(x\right\}\Rightarrow {\left(x\right]}^2=\left(x\right]+3\left(x\right\}\Rightarrow \left(x\right\}=\frac{{\left(x\right]}^2-\left(x\right]}{3}\because 0\le \left(x\right\}<1\Rightarrow 0\le \frac{{\left(x\right]}^2-\left(x\right]}{3}<1\Rightarrow 0\le {\left(x\right]}^2-\left(x\right]<3\Rightarrow {\left(x\right]}^2-\left(x\right]-3<0\&{\left(x\right]}^2-\left(x\right]>0\Rightarrow \left(x\right]\in (\frac{1-\sqrt{13}}{2},0]\cup [1,\frac{1+\sqrt{13}}{2})\Rightarrow \left(x\right]=-1,0,1,2\Rightarrow \left(x\right\}=\frac{2}{3},0,0,\frac{2}{3}\Rightarrow x=-\frac{1}{3},0,1,\frac{8}{3}$