Question

The equation $x^2-4 x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has :

• A. no solution
• B. exactly two solutions in $(-\infty, \infty)$
• C. a unique solution in $(-\infty, 1)$
• D. a unique solution in $(-\infty, \infty)$

Answer 1

Answer: (D)

$ x ^2-4 x +[ x ]+3= x [ x ] $
$\Rightarrow x ^2-4 x +3= x [ x ]-[ x ] $
$ \Rightarrow( x -1)( x -3)=[ x ] .( x -1)$
$ \Rightarrow x =1 \text { or } x -3=[ x ]$
$ \Rightarrow x -[ x ]=3$
$ \Rightarrow\{ x \}=3 \text { (Not Possible) }$
Only one solution $x=1$ in $(-\infty, \infty)$