The number of real roots of the equation `x^(2)- 3 |x| + 2 = 0` is :

Options

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

Answer

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 1: Define the cases for \( |x| \)

The absolute value function \( |x| \) can be defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

### Step 2: Case 1 - When \( x \geq 0 \)

In this case, we replace \( |x| \) with \( x \):
\[ x^2 - 3x + 2 = 0 \]
Now, we will factor the quadratic equation:
\[ (x - 1)(x - 2) = 0 \]
This gives us the roots:
\[ x = 1 \quad \text{and} \quad x = 2 \]
Both roots \( x = 1 \) and \( x = 2 \) are valid since they are non-negative.

### Step 3: Case 2 - When \( x < 0 \)

In this case, we replace \( |x| \) with \( -x \):
\[ x^2 + 3x + 2 = 0 \]
Now, we will factor this quadratic equation:
\[ (x + 1)(x + 2) = 0 \]
This gives us the roots:
\[ x = -1 \quad \text{and} \quad x = -2 \]
Both roots \( x = -1 \) and \( x = -2 \) are valid since they are negative.

### Step 4: Summary of Roots

From both cases, we have found the following real roots:
1. From Case 1: \( x = 1, 2 \)
2. From Case 2: \( x = -1, -2 \)

Thus, the total number of real roots of the equation \( x^2 - 3|x| + 2 = 0 \) is **4**.

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

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