How many real roots of the equations `x^(2) + 3|x| +2 = 0` have ?

Options

A. zero
B. one
C. two
D. Four

Answer

To solve the equation \( x^2 + 3|x| + 2 = 0 \) and determine how many real roots it has, we will consider two cases based on the definition of the absolute value function.

### Step 1: Identify Cases Based on Absolute Value
The absolute value function \( |x| \) can be expressed as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

### Step 2: Case 1: \( x \geq 0 \)
In this case, we substitute \( |x| \) with \( x \):
\[ x^2 + 3x + 2 = 0 \]
Now, we will factor the quadratic equation:
\[ (x + 1)(x + 2) = 0 \]
Setting each factor to zero gives:
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
However, since we are in the case where \( x \geq 0 \), neither \( x = -1 \) nor \( x = -2 \) are valid solutions. Therefore, there are **no real roots** in this case.

### Step 3: Case 2: \( x < 0 \)
In this case, we substitute \( |x| \) with \( -x \):
\[ x^2 - 3x + 2 = 0 \]
Now, we will factor this quadratic equation:
\[ (x - 1)(x - 2) = 0 \]
Setting each factor to zero gives:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
However, since we are in the case where \( x < 0 \), neither \( x = 1 \) nor \( x = 2 \) are valid solutions. Therefore, there are **no real roots** in this case as well.

### Conclusion
Since both cases yield no valid solutions, we conclude that the equation \( x^2 + 3|x| + 2 = 0 \) has **zero real roots**.

### Final Answer
The number of real roots of the equation is **0**.

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