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

Options

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

Answer

To find the number of real roots of the equation \( |x|^2 - 3|x| + 2 = 0 \), we will analyze the equation based on the definition of absolute value.

### Step 1: Rewrite the equation
The given equation is:
\[ |x|^2 - 3|x| + 2 = 0 \]
Let \( y = |x| \). The equation can be rewritten as:
\[ y^2 - 3y + 2 = 0 \]

### Step 2: Factor the quadratic equation
Next, we will factor the quadratic equation:
\[ y^2 - 3y + 2 = (y - 1)(y - 2) = 0 \]
This gives us the solutions:
\[ y - 1 = 0 \quad \text{or} \quad y - 2 = 0 \]
Thus, we have:
\[ y = 1 \quad \text{or} \quad y = 2 \]

### Step 3: Convert back to \( x \)
Since \( y = |x| \), we have two cases for each solution:
1. For \( y = 1 \):
- \( |x| = 1 \) gives \( x = 1 \) or \( x = -1 \)
2. For \( y = 2 \):
- \( |x| = 2 \) gives \( x = 2 \) or \( x = -2 \)

### Step 4: List all the real roots
The real roots of the original equation are:
- From \( y = 1 \): \( x = 1, -1 \)
- From \( y = 2 \): \( x = 2, -2 \)

Thus, the real roots are:
\[ x = 1, -1, 2, -2 \]

### Step 5: Count the number of real roots
In total, we have 4 real roots: \( 1, -1, 2, -2 \).

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

---