For the equation `|x^(2)| + |x| -6=0`, the roots are

Options

A. Real and equal
B. Real with sum 0
C. Real with sum 1
D. Real with product 0

Answer

To solve the equation \( |x^2| + |x| - 6 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ |x^2| + |x| - 6 = 0 \]
Since \( x^2 \) is always non-negative, we can simplify \( |x^2| \) to \( x^2 \). Thus, the equation becomes:
\[ x^2 + |x| - 6 = 0 \]

### Step 2: Consider cases for \( |x| \)
The absolute value \( |x| \) can be expressed in two cases:
1. Case 1: \( x \geq 0 \) (then \( |x| = x \))
2. Case 2: \( x < 0 \) (then \( |x| = -x \))

### Step 3: Solve Case 1: \( x \geq 0 \)
In this case, the equation becomes:
\[ x^2 + x - 6 = 0 \]
Now, we can factor this quadratic equation:
\[ (x - 2)(x + 3) = 0 \]
Setting each factor to zero gives us:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
Since we are in the case where \( x \geq 0 \), we only accept \( x = 2 \).

### Step 4: Solve Case 2: \( x < 0 \)
In this case, the equation becomes:
\[ x^2 - x - 6 = 0 \]
Again, we can factor this quadratic equation:
\[ (x - 3)(x + 2) = 0 \]
Setting each factor to zero gives us:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
Since we are in the case where \( x < 0 \), we only accept \( x = -2 \).

### Step 5: Collect the roots
From both cases, we have found the roots:
1. From Case 1: \( x = 2 \)
2. From Case 2: \( x = -2 \)

Thus, the roots of the equation \( |x^2| + |x| - 6 = 0 \) are:
\[ \boxed{2 \text{ and } -2} \]