The number of integer solutions of equation 2|x|(x^2+1)=5x^2 is

Question

UPSC Quizroom · Accepted Answer

Step 1: Check x = 0.
LHS = 2|0|(0² + 1) = 0
RHS = 5·0² = 0
So x = 0 is a solution.

Step 2: For x ≠ 0, let t = |x| > 0.
Then x² = t² and the equation becomes:
2t(t² + 1) = 5t²

Divide both sides by t (t > 0):
2(t² + 1) = 5t
→ 2t² − 5t + 2 = 0

Step 3: Solve the quadratic 2t² − 5t + 2 = 0.
Discriminant = 25 − 16 = 9
Roots: t = (5 ± 3) / 4 → t = 2 or t = 1/2

Only t = 2 is a positive integer.

Step 4: t = |x| = 2 ⇒ x = 2 or x = −2

Combine with x = 0

Integer solutions: −2, 0, 2

Final answer: 3 integer solutions

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