The equation x2-4 x+[x]+3=x[x], where [x] denotes the greatest integer function, has :

Question

The equation x2-4 x+[x]+3=x[x], where [x] denotes the greatest integer function, has : (A) no solution (B) exactly two solutions in (-∞, ∞) (C) a unique solution in (-∞, 1) (D) a unique solution in (-∞, ∞)

Tardigrade Admin · Accepted Answer

x 2-4 x +[ x ]+3= x [ x ] ⇒ x 2-4 x +3= x [ x ]-[ x ] ⇒( x -1)( x -3)=[ x ] .( x -1) ⇒ x =1 text or x -3=[ x ] ⇒ x -[ x ]=3 ⇒ x =3 text (Not Possible) Only one solution x=1 in (-∞, ∞)

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

no solution

exactly two solutions in $(- \infty, \infty)$

a unique solution in $(- \infty, 1)$

a unique solution in $(- \infty, \infty)$

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

Q. The equation x2−4x+[x]+3=x[x], where [x] denotes the greatest integer function, has :

no solution

exactly two solutions in (−∞,∞)

a unique solution in (−∞,1)

a unique solution in (−∞,∞)

x2−4x+[x]+3=x[x] ⇒x2−4x+3=x[x]−[x] ⇒(x−1)(x−3)=[x].(x−1) ⇒x=1 or x−3=[x] ⇒x−[x]=3 ⇒{x}=3 (Not Possible) Only one solution x=1 in (−∞,∞)

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

exactly two solutions in $(- \infty, \infty)$

a unique solution in $(- \infty, 1)$

a unique solution in $(- \infty, \infty)$

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