Question

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

Answer 1

Let $|x| = k$. 

Then the given equation becomes: 
$2k(k^2 + 1) = 5k^2$

Expanding both sides: 
$2k^3 + 2k = 5k^2$

Bringing all terms to one side: 
$2k^3 - 5k^2 + 2k = 0$

Factor out $k$: 
$k(2k^2 - 5k + 2) = 0$

Now solve the quadratic: 
$2k^2 - 5k + 2 = 0$

Using factorization: 
$2k^2 - 4k - k + 2 = 0$
$2k(k - 2) -1(k - 2) = 0$
$(2k - 1)(k - 2) = 0$

So the solutions are: 
$k = 0$, $k = \dfrac{1}{2}$, or $k = 2$

Since $k = |x|$, we have: 
$|x| = 0 \Rightarrow x = 0$
$|x| = \dfrac{1}{2} \Rightarrow x = \pm \dfrac{1}{2}$
$|x| = 2 \Rightarrow x = \pm 2$

So the possible values of $x$ are: 
$x = 0,\ \dfrac{1}{2},\ -\dfrac{1}{2},\ 2,\ -2$

Among these, the integral (whole number) values are: 
$x = 0,\ 2,\ -2$

∴ There are 3 integral solutions to the equation $2|x|(x^2+1) = 5x^2$.