Find the number of solutions of the equation: x^2 - 2 - 2[x] = 0 where [.] denotes the Greatest Integer Function.

Plz explain the answer.

Thanks

Since [x] , -2 and 0 (RHS) all r integers , x^2 is an integer, so x is an integer.



So, u can replace [x] by x. But when u solve now, u wont get any integral solns for x contradicting what we got earlier.



So thr r no solns.



Number of solns is zero.

I think the ans is one solution only which is x=.

Let [x]=n,an integer then nx<n+1 and

x²=2+2[x]x= [n,n+1).Also,2n+20n-1.

iff n²/2n+1<(n+1)²/2.

Now,2(n+1)<(n+1)² n²>1n>1 only as n can't be less than -1.

n² 2n+21-rt3n1+rt31<n1+rt3 from above condition.

But n is an integer and the only integer in the above interval is 2.So,n=2 and x²=6x=.

look we have <1.............1)

nd >0................................2)

and thru' the eqn.1) and 2) we have = - ........................(box(x) =)

and frm we have   lies in the intersection of (-1,-inf) U (1,inf) and (-0.268, 2.732)...(solving the inequalities)

-0.268 = 1- root(3)/2 and 2.732 = 1+root(3)/2......

but  can only take integral values.....so it can be generalised as intersection of (-1,-inf) U (1,inf) and [0,2]

(0 and 2 being the nearest integers to the limits and lying in the range also)

so seeing this we have the range as (1,2] and again as we need only integral values we take =2

and using which we get frm the origional equation.....

= root(2(1+2)) -

and  +  =root(6)

but as  + = x itself...

we have x =

rate if useful.......