Solve the system `x^(2)-2|x|=0`

Explanation

To solve the equation \( x^2 - 2|x| = 0 \), we will consider two cases based on the definition of the absolute value function.

### Step 1: Split the equation into two cases based on the sign of \( x \).

1. **Case 1:** \( x \geq 0 \)
- In this case, \( |x| = x \).
- The equation becomes:
\[ x^2 - 2x = 0 \]

2. **Case 2:** \( x < 0 \)
- In this case, \( |x| = -x \).
- The equation becomes:
\[ x^2 + 2x = 0 \]

### Step 2: Solve Case 1 (\( x \geq 0 \))

For the equation \( x^2 - 2x = 0 \):
- Factor out \( x \):
\[ x(x - 2) = 0 \]
- Set each factor to zero:
- \( x = 0 \)
- \( x - 2 = 0 \) ⇒ \( x = 2 \)

Thus, the solutions from Case 1 are \( x = 0 \) and \( x = 2 \).

### Step 3: Solve Case 2 (\( x < 0 \))

For the equation \( x^2 + 2x = 0 \):
- Factor out \( x \):
\[ x(x + 2) = 0 \]
- Set each factor to zero:
- \( x = 0 \) (not valid since \( x < 0 \))
- \( x + 2 = 0 \) ⇒ \( x = -2 \)

Thus, the solution from Case 2 is \( x = -2 \).

### Step 4: Combine all solutions

From both cases, we have the solutions:
- From Case 1: \( x = 0 \) and \( x = 2 \)
- From Case 2: \( x = -2 \)

Therefore, the complete solution set is:
\[ x = 0, \quad x = 2, \quad x = -2 \]

### Final Answer:
The solutions to the equation \( x^2 - 2|x| = 0 \) are \( x = 0, x = 2, \text{ and } x = -2 \).

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