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 number of solutions of |[x]-2x|=4, "where" [x]] is the greatest integer less than or equal to x, is

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

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

int_(0)^(pi//2)(x-[cosx])dx=. . . . . where [t] = greatest integer less or equal to t

f(x)= 1/sqrt([x]+x) , where [*] denotes the greatest integeral function less than or equals to x. Then, find the domain of f(x).

If f is an even function, then find the realvalues of x satisfying the equation f(x)=f((x+1)/(x+2))

int_(0)^(pi//2) (x-[sin x]) dx = ___ (where [x] = greatest integer not greater than x)

The function f(x)=[x]^2-[x^2] is discontinuous at (where [gamma] is the greatest integer less than or equal to gamma ), is discontinuous at

If f(x)=sin{(pi)/(3)[x]-x^(2)}" for "2ltxlt3 and [x] denotes the greatest integer less than or equal to x, then f'"("sqrt(pi//3)")" is equal to