Question

let[x] denotes the greatest integer function. What is the number of solutions of the eq. x^2-4x+[x]=0 in the interval [0,2]?​

Answer 1

x2−4x+[x] = 0

x2–4x=−[x]

when x ϵ [0,1)=>[x]=0

x2–4x=0

x=0,4

0 is in [0,1) so 0 is a solution.

when x ϵ [1,2)=>[x]=1

x2–4x=−1

x=(2+3–√),(2−3–√)

As both solutions are not in [1,2) so both are not solutions.

when x=2=>[x]=2

x2–4x=−2

x=(2+2–√),(2−2–√)

None of these are solutions as we get different value of x.

Result : So equation x2–4x+[x]=0 has only one solution x=0 in [0,2]