Question:medium

How many numbers of integral solutions of the equation \(2|x|(x^2+1)=5x^2 ?\)

Updated On: Jan 13, 2026
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Solution and Explanation

The following 3 cases are considered:
Case (i) :
x = 0. This is a valid solution since both the Left-Hand Side (LHS) and Right-Hand Side (RHS) are equal when x = 0.

Case (ii) :
x > 0
⇒ 2x(x2 + 1) = 5x2
⇒ 2(x2 + 1) = 5x
⇒ 2x2 -5x + 2 = 0
⇒ 2x2 -4x - x + 2 = 0
⇒ 2x(x - 2) -1(x -2) = 0
⇒ (x - 2)(2x - 1) = 0
Therefore, x = 2 or x = \( \frac{1}{2} \). (One integer solution exists in this case.)

Case (iii) :
x < 0
⇒ -2x(x2 + 1) = 5x2
⇒ 2x2 + 5x + 2 = 0
⇒ 2x2 + 4x + x + 2 = 0
⇒ 2x(x + 2) + 1(x + 2) = 0
⇒ (x + 2)(2x + 1) = 0
Therefore, x = -2 or x = \( -\frac{1}{2} \). (One integer solution exists in this case.)

Consequently, the total number of integer solutions is 3, specifically 0, 2, and -2.

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