How many real roots does the quadratic equation `f(x) = x^(2) + 3 |x|+2 = 0` have ?

Options

A. One
B. Two
C. Fore
D. No real root

Explanation

To determine how many real roots the quadratic equation \( f(x) = x^2 + 3|x| + 2 = 0 \) has, we need to analyze the equation by considering the behavior of the absolute value function.

### Step-by-Step Solution:

1. **Understand the Absolute Value**:
The absolute value function \( |x| \) behaves differently for positive and negative values of \( x \). Therefore, we will consider two cases:
- Case 1: \( x \geq 0 \) (where \( |x| = x \))
- Case 2: \( x < 0 \) (where \( |x| = -x \))

2. **Case 1: \( x \geq 0 \)**:
In this case, the equation becomes:
\[ f(x) = x^2 + 3x + 2 = 0 \]
We can factor this quadratic equation:
\[ (x + 1)(x + 2) = 0 \]
Setting each factor to zero gives us the roots:
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
However, since we are in the case where \( x \geq 0 \), these roots \( -1 \) and \( -2 \) are not valid.

3. **Case 2: \( x < 0 \)**:
In this case, the equation becomes:
\[ f(x) = x^2 - 3x + 2 = 0 \]
We can factor this quadratic equation as well:
\[ (x - 1)(x - 2) = 0 \]
Setting each factor to zero gives us the roots:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
However, since we are in the case where \( x < 0 \), these roots \( 1 \) and \( 2 \) are not valid.

4. **Conclusion**:
Since both cases yielded roots that are not valid in their respective domains, we conclude that the quadratic equation \( f(x) = x^2 + 3|x| + 2 = 0 \) has no real roots.

### Final Answer:
The quadratic equation has **no real roots**.