If `x=4/9` satisfies the equation `(log)_a(x^2-x+2)>(log)_a(-x^2+2x+3),` then the sum of all possible distinct values of `[x]` is (where`[dot]` represetns the greatest integer function) ___

Answer

2

Explanation

`log_(a)(x^(2)-x+2) gt log_(a)(-x^(2)+2x+3)`
Putting `x=(4)/(9),log_(a)((142)/(81)) gt log_(a)((299)/(81))`
` :' (142)/(81) lt (299)/(81)," we have " 0 lt a lt 1`
` :. log_(a) (x^(2)-x+2) gt log_(a)(-x^(2)+2x+3)`
gives ` 0 lt x^(2)-x+2 lt -x^(2) +2x+3`
`or x^(2)-x+2 gt 0 and 2x^(2)-3x-1 lt 0`
`or (3-sqrt(17))/(4) lt x lt (3+sqrt(17))/(4)`