Find all the values of x that satisfy `x^2-3|x|+2=0`

Options

A. a) 0
B. b) 2
C. c) 4
D. d) none of these

Answer

To solve the equation \( x^2 - 3|x| + 2 = 0 \), we need to consider two cases for \( |x| \) since it can take on different values depending on whether \( x \) is positive or negative.

### Step 1: Consider the case when \( x \geq 0 \)
In this case, \( |x| = x \). The equation becomes:
\[ x^2 - 3x + 2 = 0 \]

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

### Step 3: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 1 = 0 \) → \( x = 1 \)
2. \( x - 2 = 0 \) → \( x = 2 \)

Since both solutions \( x = 1 \) and \( x = 2 \) are non-negative, they are valid solutions for this case.

### Step 4: Consider the case when \( x < 0 \)
In this case, \( |x| = -x \). The equation becomes:
\[ x^2 + 3x + 2 = 0 \]

### Step 5: Factor the quadratic equation
We can factor this quadratic as well:
\[ x^2 + 3x + 2 = (x + 1)(x + 2) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 1 = 0 \) → \( x = -1 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

Since both solutions \( x = -1 \) and \( x = -2 \) are negative, they are valid solutions for this case.

### Final Solutions
Combining the solutions from both cases, we have:
\[ x = 1, \quad x = 2, \quad x = -1, \quad x = -2 \]

### Summary of Solutions
The values of \( x \) that satisfy the equation \( x^2 - 3|x| + 2 = 0 \) are:
\[ x = 1, \quad x = 2, \quad x = -1, \quad x = -2 \]