The equation x 2 - 4 x + [ x ] + 3 = x [ x ] , where [x] denotes the greatest integer function, has [2023]

Question

The equation $x^{2} - 4 x + [x] + 3 = x [x],$ where [x] denotes the greatest integer function, has [2023]

Smart Achievers · Accepted Answer

(4)

$x^{2} - 4 x + [x] + 3 = x [x]$

$\Rightarrow x^{2} - x [x] - (x - [x]) + 3 (x - 1) = 0$

$\Rightarrow x (x - [x]) - 1 (x - [x]) - 3 (x - 1) = 0$

$\Rightarrow (x - 1) (x - [x]) - 3 (x - 1) = 0$

$\Rightarrow (x - 1) (x - [x] - 3) = 0$

$\Rightarrow (x - 1) ({x} - 3) = 0 \Rightarrow x = 1 as {x} \neq 3$

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