X²-2x²/|x|=0

To solve the equation \( x^2 - \frac{2x^2}{|x|} = 0 \), we can first simplify it. ### Step 1: Simplify the equation The expression can be rewritten as follows: \[ x^2 - 2 \frac{x^2}{|x|} = 0 \] This can be simplified to: \[ x^2 - \frac{2x^2}{|x|} = 0 \] ### Step 2: Analyze the absolute value The absolute value function \( |x| \) behaves differently based on whether \( x \) is positive or negative. 1. **Case 1: \( x > 0 \)** - Here, \( |x| = x \). - Substitute into the equation: \[ x^2 - \frac{2x^2}{x} = x^2 - 2x = 0 \] - Factoring out \( x \): \[ x(x - 2) = 0 \] - This gives the solutions \( x = 0 \) or \( x = 2 \). However, since we are in the case where \( x > 0 \), we discard \( x = 0 \) and have \( x = 2 \). 2. **Case 2: \( x < 0 \)** - Here, \( |x| = -x \). - Substitute into the equation: \[ x^2 - \frac{2x^2}{-x} = x^2 + 2x = 0 \] - Factoring out \( x \): \[ x(x + 2) = 0 \] - This gives the solutions \( x = 0 \) or \( x = -2 \). However, since we are in the case where \( x < 0 \), we discard \( x = 0 \) and have \( x = -2 \). ### Final Solutions The solutions to the equation \( x^2 - \frac{2x^2}{|x|} = 0 \) are: \[ x = 2 \quad \text{and} \quad x = -2 \]