The number of real solutions of the equation `|x|^(2)-3|x|+2=0` is :

Options

A. 4
B. 1
C. 3
D. 2

Answer

To solve the equation \( |x|^2 - 3|x| + 2 = 0 \) and find the number of real solutions, we can follow these steps:

### Step-by-Step Solution:

1. **Substitute \( |x| \)**:
Let \( y = |x| \). Then the equation becomes:
\[ y^2 - 3y + 2 = 0 \]

2. **Factor the Quadratic Equation**:
We can factor the quadratic equation:
\[ (y - 1)(y - 2) = 0 \]

3. **Find the Values of \( y \)**:
Setting each factor to zero gives us:
\[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \]
\[ y - 2 = 0 \quad \Rightarrow \quad y = 2 \]

4. **Convert Back to \( x \)**:
Since \( y = |x| \), we have:
- For \( y = 1 \):
\[ |x| = 1 \quad \Rightarrow \quad x = 1 \quad \text{or} \quad x = -1 \]
- For \( y = 2 \):
\[ |x| = 2 \quad \Rightarrow \quad x = 2 \quad \text{or} \quad x = -2 \]

5. **Count the Total Solutions**:
The values of \( x \) we found are:
- \( x = 1 \)
- \( x = -1 \)
- \( x = 2 \)
- \( x = -2 \)

Thus, we have a total of 4 real solutions.

### Conclusion:
The number of real solutions of the equation \( |x|^2 - 3|x| + 2 = 0 \) is **4**.