The number of real roots of the equation` x^2-3 `modx` +2=0`` is

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Identify Cases Based on Absolute Value**:
The absolute value function \( |x| \) behaves differently based on whether \( x \) is non-negative or negative. Therefore, we will consider two cases:
- Case 1: \( x \geq 0 \)
- Case 2: \( x < 0 \)

2. **Case 1: \( x \geq 0 \)**:
In this case, \( |x| = x \). Thus, the equation becomes:
\[ x^2 - 3x + 2 = 0 \]
We can factor this quadratic equation:
\[ (x - 1)(x - 2) = 0 \]
Setting each factor to zero gives us the roots:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
Both roots \( x = 1 \) and \( x = 2 \) are non-negative, so we have **two real roots from this case**.

3. **Case 2: \( x < 0 \)**:
In this case, \( |x| = -x \). Thus, the equation becomes:
\[ x^2 - 3(-x) + 2 = 0 \quad \Rightarrow \quad x^2 + 3x + 2 = 0 \]
We can factor this quadratic equation:
\[ (x + 1)(x + 2) = 0 \]
Setting each factor to zero gives us the roots:
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
Both roots \( x = -1 \) and \( x = -2 \) are negative, so we have **two real roots from this case**.

4. **Total Number of Real Roots**:
From both cases, we have:
- 2 real roots from Case 1 (non-negative)
- 2 real roots from Case 2 (negative)

Therefore, the total number of real roots of the equation \( x^2 - 3 |x| + 2 = 0 \) is:
\[ 2 + 2 = 4 \]

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