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

Options

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

Answer

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

### Step 1: Rewrite the Equation
The equation can be rewritten as:
\[ |x^2| - 3|x| + 2 = 0 \]
Since \( |x^2| = x^2 \) for all real \( x \), we can simplify this to:
\[ x^2 - 3|x| + 2 = 0 \]

### Step 2: Substitute \( |x| \)
Let \( y = |x| \). The equation now becomes:
\[ y^2 - 3y + 2 = 0 \]

### Step 3: Factor the Quadratic
Next, we need to factor the quadratic equation:
\[ y^2 - 3y + 2 = (y - 1)(y - 2) = 0 \]

### Step 4: Solve for \( y \)
Setting each factor to zero gives us:
\[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \]
\[ y - 2 = 0 \quad \Rightarrow \quad y = 2 \]

### Step 5: Convert Back to \( x \)
Since \( y = |x| \), we have:
1. \( |x| = 1 \) which gives \( x = 1 \) or \( x = -1 \)
2. \( |x| = 2 \) which gives \( x = 2 \) or \( x = -2 \)

### Step 6: Count the Solutions
The solutions for \( x \) are:
- \( x = 1 \)
- \( x = -1 \)
- \( x = 2 \)
- \( x = -2 \)

Thus, there are a total of 4 solutions.

### Final Answer
Therefore, the number of solutions to the equation \( |x^2| - 3|x| + 2 = 0 \) is:
\[ \text{Total number of solutions} = 4 \]

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