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)

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

Answer 2

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 (−∞,∞)