The number of real solutions of the equation `|x^(2)|-3|x|+2=0` is (a) 3 (b) 4 (c) 1 (d) 3.

Answer

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

### Step 1: Rewrite the equation
Since \( |x^2| = x^2 \) for all real \( x \), we can rewrite the equation as:
\[ x^2 - 3|x| + 2 = 0 \]

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

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

### 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: Find values of \( x \)
Since \( y = |x| \), we have:
1. For \( y = 1 \):
- \( |x| = 1 \) leads to \( x = 1 \) or \( x = -1 \)

2. For \( y = 2 \):
- \( |x| = 2 \) leads to \( x = 2 \) or \( x = -2 \)

### Step 6: Count the 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.

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

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