Number of integers stastifying the equation `|x^(2)+5x|+| x-x^(2)|=|6x|` is:

Options

A. 3
B. 5
C. 7
D. 9

Answer

To solve the equation \( |x^2 + 5x| + |x - x^2| = |6x| \), we will follow these steps:

### Step 1: Define the expressions
Let:
- \( a = x^2 + 5x \)
- \( b = x - x^2 \)

The equation can be rewritten as:
\[ |a| + |b| = |6x| \]

### Step 2: Analyze the conditions for the absolute values
The absolute value expressions depend on the signs of \( a \) and \( b \). We need to consider the cases when \( a \) and \( b \) are non-negative or negative.

### Step 3: Find the critical points
1. **For \( a = x^2 + 5x \)**:
- Set \( a = 0 \):
\[ x^2 + 5x = 0 \implies x(x + 5) = 0 \implies x = 0 \text{ or } x = -5 \]

2. **For \( b = x - x^2 \)**:
- Set \( b = 0 \):
\[ x - x^2 = 0 \implies x(1 - x) = 0 \implies x = 0 \text{ or } x = 1 \]

### Step 4: Determine the intervals
The critical points are \( -5, 0, 1 \). We will analyze the intervals:
1. \( (-\infty, -5) \)
2. \( [-5, 0) \)
3. \( [0, 1) \)
4. \( [1, \infty) \)

### Step 5: Analyze each interval
1. **Interval \( (-\infty, -5) \)**:
- \( a < 0 \) and \( b < 0 \)
- Equation becomes:
\[ -a - b = -6x \implies - (x^2 + 5x) - (x - x^2) = -6x \]
Simplifying gives:
\[ -x^2 - 5x - x + x^2 = -6x \implies -5x = -6x \implies x = 0 \text{ (not in this interval)} \]

2. **Interval \( [-5, 0) \)**:
- \( a \geq 0 \) and \( b < 0 \)
- Equation becomes:
\[ a - b = -6x \implies (x^2 + 5x) - (x - x^2) = -6x \]
Simplifying gives:
\[ x^2 + 5x - x + x^2 = -6x \implies 2x^2 + 12x = 0 \implies 2x(x + 6) = 0 \]
Solutions are \( x = 0 \) or \( x = -6 \) (only \( -6 \) is valid in this interval).

3. **Interval \( [0, 1) \)**:
- \( a \geq 0 \) and \( b \geq 0 \)
- Equation becomes:
\[ a + b = 6x \implies (x^2 + 5x) + (x - x^2) = 6x \]
Simplifying gives:
\[ 5x + x = 6x \implies 6x = 6x \text{ (true for all \( x \) in this interval)} \]
Valid integers are \( 0 \) and \( 1 \).

4. **Interval \( [1, \infty) \)**:
- \( a \geq 0 \) and \( b < 0 \)
- Equation becomes:
\[ a - b = 6x \implies (x^2 + 5x) - (x - x^2) = 6x \]
Simplifying gives:
\[ 2x^2 + 4x = 0 \implies 2x(x + 2) = 0 \]
Solutions are \( x = 0 \) or \( x = -2 \) (not valid in this interval).

### Step 6: Collect valid integer solutions
From the intervals:
- From \( [-5, 0) \): valid integer is \( -6 \).
- From \( [0, 1) \): valid integers are \( 0 \) and \( 1 \).

### Conclusion
The valid integers satisfying the equation are:
- \( -5, -4, -3, -2, -1, 0, 1 \)

Thus, the total number of integers is **7**.