[x]^2= x + 2{x} Solve for x.

Question

[.] is greatest integer function of x and {.} is Fractional part of x. Solve for x. Has 4 solutions.

Answer 1

Any real number can be expressed as the sum of its integer part and its fractional part, i.e.

x = [x]+{x}

 

Substituting {x} = x - [x] into [x]² = x + 2{x} yields

[x]² = 3x - 2[x]

[x]² +2[x]  = 3x

[x]² +2[x] +1 = 3x +1. Adding 1 to both sides to complete the square of the left hand side (LHS)

[x] = SQRT(3x +1) - 1

 

Since SQRT(3x +1)≥ 0 and [x] must be an integer, the solution x can be found by

letting 3x +1 equal to 0, 1, 4 or 9 and solve for x.

 

The solutions are -1/3, 0, 1 and 8/3.