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:

\[ x = 2 \quad \text{and} \quad x = -2 \]