X²+4x²/|x|=0

To solve the equation \( x^2 + \frac{4x^2}{|x|} = 0 \), let's first rewrite the equation for clarity: \[ x^2 + \frac{4x^2}{|x|} = 0 \] Next, we can identify two cases based on the value of \( x \) since the absolute value function, \( |x| \), behaves differently for positive and negative values of \( x \). ### Case 1: \( x > 0 \) In this case, \( |x| = x \). Substituting this into the equation gives: \[ x^2 + \frac{4x^2}{x} = 0 \] This simplifies to: \[ x^2 + 4x = 0 \] Factoring out \( x \): \[ x(x + 4) = 0 \] This results in two potential solutions: 1. \( x = 0 \) 2. \( x + 4 = 0 \Rightarrow x = -4 \) (but this does not apply since \( x > 0 \)) Thus, no solutions come from Case 1. ### Case 2: \( x < 0 \) In this case, \( |x| = -x \). Substituting this into the equation gives: \[ x^2 + \frac{4x^2}{-x} = 0 \] This simplifies to: \[ x^2 - 4x = 0 \] Factoring out \( x \): \[ x(x - 4) = 0 \] This results in two potential solutions: 1. \( x = 0 \) 2. \( x - 4 = 0 \Rightarrow x = 4 \) (but this does not apply since \( x < 0 \)) Thus, again, no solutions come from Case 2. ### Conclusion Since neither case yields a valid solution other than \( x = 0 \), we can conclude that the only solution to the equation \( x^2 + \frac{4x^2}{|x|} = 0 \) is: \[ \boxed{0} \]