The number of real solutions of `x^(2) - 3|x| + 2 = 0` is

Options

A. 4
B. 2
C. 1
D. 0

Answer

To find the number of real solutions of the equation \( x^2 - 3|x| + 2 = 0 \), we need to consider two cases based on the definition of the absolute value function.

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

In this case, \( |x| = x \). Therefore, the equation becomes:
\[ x^2 - 3x + 2 = 0 \]

### Step 2: Solve the quadratic equation

We can factor the quadratic equation:
\[ (x - 1)(x - 2) = 0 \]
Setting each factor to zero gives us:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

Both solutions \( x = 1 \) and \( x = 2 \) are valid since they are non-negative.

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

In this case, \( |x| = -x \). Therefore, the equation becomes:
\[ x^2 + 3x + 2 = 0 \]

### Step 4: Solve the quadratic equation

We can factor this quadratic equation as well:
\[ (x + 1)(x + 2) = 0 \]
Setting each factor to zero gives us:
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

Both solutions \( x = -1 \) and \( x = -2 \) are valid since they are negative.

### Step 5: Conclusion

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

Thus, the total number of real solutions is:
\[ \text{Total solutions} = 4 \]

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

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