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If (x)2 = [x]2 + 2x then which of the following is true. Where [x] represents greatest integer less than or equal to x. (x) represents integer just greater than or equal to x. I represents Integer.

If (x)2 = [x]2 + 2x then which of the following is true. Where [x] represents greatest integer less than or equal to x. (x) represents integer just greater than or equal to x. I represents Integer. Correct Answer <span class="fontstyle2">x = n + 1/2, n </span><span style="color: rgb(32, 33, 36);">∈</span><span class="fontstyle2"> I or </span>x = 0

Calculation:

Case I:

⇒ Let x = n ∈ I

⇒ Given equation becomes:

⇒ n2 = n2 + 2n

⇒ n = 0

Case II:

⇒ Let x ∈ I

⇒ i.e. n < x < n + 1

⇒ Given equation becomes:

⇒ (n - 1)2 = n2 + 2x

⇒ x = n + 1/2, n ∈ I

⇒ Therefore x = 0 or x = n + 1/2; n ∈ I
+ {x}; where {x} represent fraction part of x.

⇒ x = (x) - (1 - {x})

⇒ (x + 1 - {x})2 = (x - {x})2 + 2x (Using given equation)

⇒ (x + 1 - {x})2 + 1 + 2 (x - {x})2 = (x - {x})2 + 2x

⇒ 1 - 2 {x} = 0

⇒ {x} = 1/2

⇒ x = n + 1/2, n ∈ I

⇒ Also, x = 0, by observation.

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