Question

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

Answer 1

Let's consider the following 3 cases :
Case (i) :
x = 0 , This is a solution, as both LHS and RHS will be equal when value of x is 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
So, x =2 or \(\frac{1}{2}\) , (Here 1 integer solution is there)

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
So, x = -2 or \(-\frac{1}{2}\), (Here 1 integer solution is there)

Therefore, the total number of integer solution are 0, 2, -2 i.e 3 solutions.