Solve the equation (x) 2 = [x] 2 + 2x, where [x] and (x) ar…

Maths Functions, Limits, Continuity and Differentiability General Subjective Type

Published on: August 14, 2026

Solve the equation (x) 2 = [x] 2 + 2x, where [x] and (x) are integers just less than or equal to x and just greater than or equal to x respectively.

Text Solution

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The correct answer is:

(x) = [x]; (x) = [x]

Sol. Let {x} denotes the fractional part of x.

Case- I

When 0 ≤ {x} < ; we have (x) = [x]

⇒ [x] 2 = [x] 2 + 2x ⇒ x = 0

Case- II

When ≤ {x} < 1; we have (x) = [x] + 1

∴ (x) 2 = [x] 2 + 2x ⇒ {[x] +1} 2 = [x] 2 + 2x ⇒ x = [x] +

∴ x = n + ; n ∈ I

Hence the solution set is given by x = 0, n + ; n ∈ Z

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