Sum of integral roots of the equation `|x^2-x-6|=x+2` is

Options

A. two positive roots
B. two real roots
C. three real roots
D. four real roots

Answer

To find the sum of integral roots of the equation \( |x^2 - x - 6| = x + 2 \), we will break down the problem step by step.

### Step 1: Break Down the Absolute Value
The equation involves an absolute value, which means we need to consider two cases based on the expression inside the absolute value:

1. \( x^2 - x - 6 = x + 2 \)
2. \( -(x^2 - x - 6) = x + 2 \)

### Step 2: Solve the First Case
For the first case:
\[ x^2 - x - 6 = x + 2 \]
Rearranging gives:
\[ x^2 - 2x - 8 = 0 \]
Now, we can factor this quadratic:
\[ (x - 4)(x + 2) = 0 \]
Setting each factor to zero gives us the roots:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

### Step 3: Check Validity of Roots for the First Case
Since we derived this from the case where \( x^2 - x - 6 \geq 0 \), we need to check when this holds true. The roots of \( x^2 - x - 6 = 0 \) are \( x = -2 \) and \( x = 3 \). The intervals are:
- \( x < -2 \)
- \( -2 \leq x < 3 \)
- \( x \geq 3 \)

For \( x \geq 3 \), \( x = 4 \) is valid. For \( -2 \leq x < 3 \), \( x = -2 \) is also valid.

### Step 4: Solve the Second Case
Now, for the second case:
\[ -(x^2 - x - 6) = x + 2 \]
This simplifies to:
\[ -x^2 + x + 6 = x + 2 \]
Rearranging gives:
\[ -x^2 + 4 = 0 \quad \Rightarrow \quad x^2 = 4 \]
Taking the square root of both sides, we get:
\[ x = 2 \quad \text{or} \quad x = -2 \]

### Step 5: Check Validity of Roots for the Second Case
For this case, we need \( -(x^2 - x - 6) \geq 0 \), which means \( x^2 - x - 6 \leq 0 \). This is true in the interval \( -2 \leq x \leq 3 \).

Both roots \( x = 2 \) and \( x = -2 \) are valid.

### Step 6: Collect All Valid Roots
From both cases, the valid integral roots are:
- From Case 1: \( x = 4, -2 \)
- From Case 2: \( x = 2, -2 \)

The unique integral roots are \( 4, -2, 2 \).

### Step 7: Calculate the Sum of Integral Roots
Now, we sum the unique integral roots:
\[ 4 + (-2) + 2 = 4 \]

### Final Answer
The sum of the integral roots of the equation \( |x^2 - x - 6| = x + 2 \) is \( \boxed{4} \).

---