Question

[x]^2-5 [x] +6=0 .find x if [ ] represent greatest integer function

Answer 1

given
quadratic eqn

[X]²-5[X] +6= 0

[X]²-3[X] -2[X] +6 =0

[X]( [X]-3)-2([X]-3) =0

([X] -2) ([X]-3 ) = 0

[X] = 2 .
thus X € [2,3)

[X]= 3
x€ [3,4)

so overall values of X

€ [2,3)U[3,4)

x € [2,4)

Answer 2

it is given that [x]² - 5[x] + 6 = 0 where [ . ] represent greatest integer function.

we have to find the value of x.

[x]² - 5[x] + 6 = 0

⇒ [x]² - 2[x] - 3[x] + 6 = 0

⇒[x]([x] - 2) - 3([x] - 2) = 0

⇒([x] - 3)([x] - 2) = 0

⇒[x] = 3, 2

we know, when [x] = n ⇒n - 1 < x ≤ n

so, [x] = 2 ⇒2 - 1 < x ≤ 2 ⇒1 < x ≤ 2

[x] = 3 ⇒3 - 1 < x ≤ 3 ⇒2 < x ≤ 3

hence x ∈ [2, 1) U [3, 2)