Number of real solutions of the equation `x ^(2) +3|x| + 2 =0` is:
i) 0
ii) 2
iii) 3
iv) 4

Options

A. 0
B. 2
C. 3
D. 4

Answer

To determine the number of real solutions for the equation \( x^2 + 3|x| + 2 = 0 \), we can follow these steps:

### Step 1: Analyze the equation
The equation is given as:
\[ x^2 + 3|x| + 2 = 0 \]
Here, \( |x| \) represents the absolute value of \( x \), which is always non-negative.

### Step 2: Consider cases for \( |x| \)
Since \( |x| \) can be expressed as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

We will analyze both cases separately.

### Case 1: \( x \geq 0 \)
In this case, \( |x| = x \). 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:
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \quad (\text{not valid since } x \geq 0) \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \quad (\text{not valid since } x \geq 0) \]
Thus, there are no valid solutions in this case.

### Case 2: \( x < 0 \)
In this case, \( |x| = -x \). 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:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \quad (\text{not valid since } x < 0) \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \quad (\text{not valid since } x < 0) \]
Thus, there are no valid solutions in this case either.

### Step 3: Conclusion
Since there are no valid solutions from both cases, we conclude that the number of real solutions to the equation \( x^2 + 3|x| + 2 = 0 \) is:
\[ \text{Number of real solutions} = 0 \]

### Final Answer
The correct option is:
i) 0

---