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Joint Entrance Examination
Mathematics
Quadratic Equations and Inequalities
Question 147
Question 147 of 157
Medium
MCQ
The equation \[{x^2} - 4x + [x] + 3 = x[x]\], where \[[x]\] denotes the greatest integer function, has :
A
exactly two solutions in (\[-\infty,\infty\])
B
no solution
C
a unique solution in (\[-\infty,\infty\])
D
a unique solution in (\[-\infty,1\])
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Official solution
\[{x^2} - 4x + [x] + 3 = x[x]\] \[ \Rightarrow {x^2} - 4x + [x] + 3 - x[x] = 0\] \[ \Rightarrow (x - 1)(x - 3) - [x](x - 1) = 0\] \[ \Rightarrow (x - 1)(x - [x] - 3) = 0\] \[\therefore\]
x
=
1
x = 1
x
=
1
or \[x - [x] - 3 = 0\] \[ \Rightarrow \{ x\} - 3 = 0\] [As \[\{ x\} = x - [x]\]] \[ \Rightarrow \{ x\} = 3\]
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