The number of real roots of `|x^(2)| - 5|x| + 6 = 0` is

Options

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

Answer

To solve the equation \( |x^2| - 5|x| + 6 = 0 \) and find the number of real roots, we will analyze the equation by considering the cases for the absolute values.

### Step-by-Step Solution:

1. **Understanding the Absolute Values**:
Since \( |x^2| = x^2 \) for all real \( x \) (as \( x^2 \) is always non-negative), we can rewrite the equation as:
\[ x^2 - 5|x| + 6 = 0 \]

2. **Case 1: \( x \geq 0 \)**:
In this case, \( |x| = x \). Thus, the equation becomes:
\[ x^2 - 5x + 6 = 0 \]
We can factor this quadratic equation:
\[ (x - 2)(x - 3) = 0 \]
Setting each factor to zero gives us:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
Therefore, we have two real roots from this case: \( x = 2 \) and \( x = 3 \).

3. **Case 2: \( x < 0 \)**:
Here, \( |x| = -x \). Therefore, the equation becomes:
\[ x^2 + 5x + 6 = 0 \]
We can factor this quadratic equation:
\[ (x + 2)(x + 3) = 0 \]
Setting each factor to zero gives us:
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
Thus, we have two additional real roots from this case: \( x = -2 \) and \( x = -3 \).

4. **Counting the Total Real Roots**:
From both cases, we have found four real roots:
- From Case 1: \( x = 2, 3 \)
- From Case 2: \( x = -2, -3 \)

### Conclusion:
The total number of real roots of the equation \( |x^2| - 5|x| + 6 = 0 \) is **4**.