Let x denote the greatest integer function What is the number of solutions of the equation x2 ndash ...........

Mathematics (NDA) Differential Equations

Let [x] denote the greatest integer function. What is the number of solutions of the equation x² – 4x + [x] = 0 in the interval [0, 2]?

Zero (No solution)

One

Two

Three

x²−4x+[x] = 0
x²–4x=−[x]
when x ? [0,1)=>[x]=0
x²–4x=0
x=0,4
0 is in [0,1) so 0 is a solution.

when x [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]

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