Home
Class 12
MATHS
The equation (x)^(2)=[x]^(2)+2x where [x...

The equation `(x)^(2)=[x]^(2)+2x` where `[x]` and `(x)` are the integers just less than or equal to `x` and just greater than or equal to `x` respectively, then number of values of `x` satisfying the given equation

Text Solution

Verified by Experts

Case I If `x epsilonI` then
`x=[x]=(x)`
The given equation reduces to
`x^(2)=x^(2)+2x`
`implies2x=0` or `x=0`….i
Case II If `x!inI`, then `(x)=[x]+1`
The given equation reduces to
`([x]+1)^(2)=[x]^(2)+2x`
`implies1=2(x-[x])` or `{x}=1/2`
`:.x=[x]+1/2=n+1/2,n epsilonI`............ii ,brgt Hence the solution of the original equation is `x=0, n+1/2, n epsilon I`.
Promotional Banner

Similar Questions

Explore conceptually related problems

Solve the equation x^(3)-[x]=3 , where [x] denotes the greatest integer less than or equal to x .

The funcction f(x)=x-[x]+cosx , where [x] is the greatest integer less than or equal to x, is a :

If [x] is the greatest integer less than or equal to x and (x) be the least integer greater than or equal to x and [x]^(2)+(x)^(2)gt25 then x belongs to

The equation x^2 - 2 = [sin x], where [.] denotes the greatest integer function, has

The function f(x)=[x] , where [x] denotes the greatest than x, is

The function f(x) = [x], where [x] denotes the greatest integer not greater than x, is

Find the number of integral values of x satisfying sqrt(-x^2+10x-16) lt x -2

The value of x satisfying x^2 +5x + 6 = 0 is

The number of real roots of the equation |x|^(2)-3|x|+2=0 is

The number of solutions for the equation sin2x+cos 4x=2 is