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Question: Total number of solutions of [x]<sup>2</sup> = x + 2{x}, where [.] and {.} denotes the greatest inte...

Total number of solutions of [x]2 = x + 2{x}, where [.] and {.} denotes the greatest integer function and fractional part respectively is equal to

A

2

B

4

C

6

D

None of these

Answer

4

Explanation

Solution

[x]2 = x + 2{x}

⇒ [x]2 = [x] + 3[x] ⇒ {x}=[x]2−[x]3\{ x \} = \frac { [ x ] ^ { 2 } - [ x ] } { 3 }

⇒ 0≤[x]2−[x]30 \leq \frac { [ x ] ^ { 2 } - [ x ] } { 3 } < 1 ⇒ 0 ≤ [x]2 − [x] < 3

⇒ [x] ∈ (1−132,0]∪[1,1+132)\left( \frac { 1 - \sqrt { 13 } } { 2 } , 0 \right] \cup \left[ 1 , \frac { 1 + \sqrt { 13 } } { 2 } \right)

⇒ [x] = −1, 0, 1, 2 ⇒ {x} = 23\frac { 2 } { 3 } , 0, 0, 23\frac { 2 } { 3 }

⇒ x = −13- \frac { 1 } { 3 }, 0, 1, 83\frac { 8 } { 3 }