The equation $x^2-4x+\left[x\right]+3=x\left[x\right]$ where [x] denote the greatest integer function, has

x³ - 4x + 3 = x[x] - [x]
(x - 1)(x - 3) = (x - 1)[x] $\Rightarrow$x = 1 =0 or x - 3 = [x]
$\Rightarrow$x - [x] = 3 
$\Rightarrow${x} = 3 not possible
$\therefore$x = 1 is only solution