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

Explanation

To solve the equation \( (x)^2 = [x]^2 + 2x \), where \( [x] \) is the greatest integer less than or equal to \( x \) and \( (x) \) is the smallest integer greater than or equal to \( x \), we will analyze two cases: when \( x \) is an integer and when \( x \) is not an integer.

### Step 1: Case 1 - When \( x \) is an integer

If \( x \) is an integer, then both \( [x] \) and \( (x) \) equal \( x \). Therefore, we can rewrite the equation as:

\[
x^2 = x^2 + 2x
...