If `|x^(2)|-3|x|+2=0` then x =

Options

A. 1 or 2
B. `-1` or `-2`
C. `+-1` or `+-2`
D. `+-3`

Answer

To solve 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 now becomes:
\[ y^2 - 3y + 2 = 0 \]

### Step 3: Factor the quadratic equation
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 can find \( x \) for each value of \( y \):
1. If \( y = 1 \):
\[ |x| = 1 \quad \Rightarrow \quad x = 1 \text{ or } x = -1 \]
2. If \( y = 2 \):
\[ |x| = 2 \quad \Rightarrow \quad x = 2 \text{ or } x = -2 \]

### Step 6: Combine all solutions
Thus, the complete set of solutions for \( x \) is:
\[ x = 1, -1, 2, -2 \]

### Final Answer
The values of \( x \) that satisfy the equation are:
\[ x = \pm 1 \text{ and } \pm 2 \]