If the inequality `x^(2)+ax+a^(2)+6alt0` is satisfied for all `x" in "(1, 2)`, then the sum of all the integral values of a must be equal to

Options

A. `-10`
B. `-15`
C. `-21`
D. `-28`

Answer

To solve the inequality \( x^2 + ax + a^2 + 6a < 0 \) for all \( x \) in the interval \( (1, 2) \), we need to analyze the quadratic expression and find the values of \( a \) that satisfy the conditions.

### Step 1: Evaluate the inequality at the endpoints of the interval

1. **Evaluate at \( x = 1 \)**:
\[ f(1) = 1^2 + a(1) + a^2 + 6a < 0 \]
Simplifying this gives:
\[ 1 + a + a^2 + 6a < 0 \implies a^2 + 7a + 1 < 0 \]

2. **Evaluate at \( x = 2 \)**:
\[ f(2) = 2^2 + a(2) + a^2 + 6a < 0 \]
Simplifying this gives:
\[ 4 + 2a + a^2 + 6a < 0 \implies a^2 + 8a + 4 < 0 \]

### Step 2: Analyze the quadratic inequalities

1. **For \( a^2 + 7a + 1 < 0 \)**:
- Calculate the discriminant:
\[ D = b^2 - 4ac = 7^2 - 4 \cdot 1 \cdot 1 = 49 - 4 = 45 \]
- The roots are:
\[ a = \frac{-7 \pm \sqrt{45}}{2} = \frac{-7 \pm 3\sqrt{5}}{2} \]
- The quadratic \( a^2 + 7a + 1 < 0 \) is satisfied between the roots:
\[ a \in \left( \frac{-7 - 3\sqrt{5}}{2}, \frac{-7 + 3\sqrt{5}}{2} \right) \]

2. **For \( a^2 + 8a + 4 < 0 \)**:
- Calculate the discriminant:
\[ D = 8^2 - 4 \cdot 1 \cdot 4 = 64 - 16 = 48 \]
- The roots are:
\[ a = \frac{-8 \pm \sqrt{48}}{2} = \frac{-8 \pm 4\sqrt{3}}{2} = -4 \pm 2\sqrt{3} \]
- The quadratic \( a^2 + 8a + 4 < 0 \) is satisfied between the roots:
\[ a \in \left( -4 - 2\sqrt{3}, -4 + 2\sqrt{3} \right) \]

### Step 3: Find the intersection of the intervals

Now we need to find the intersection of the intervals obtained from the two inequalities:

1. **Interval from \( a^2 + 7a + 1 < 0 \)**:
\[ \left( \frac{-7 - 3\sqrt{5}}{2}, \frac{-7 + 3\sqrt{5}}{2} \right) \]

2. **Interval from \( a^2 + 8a + 4 < 0 \)**:
\[ \left( -4 - 2\sqrt{3}, -4 + 2\sqrt{3} \right) \]

### Step 4: Determine the integral values of \( a \)

We need to find the integral values of \( a \) that lie within the intersection of the two intervals.

1. **Calculate approximate values**:
- \( \sqrt{5} \approx 2.236 \) gives \( \frac{-7 - 3\sqrt{5}}{2} \approx -8.354 \) and \( \frac{-7 + 3\sqrt{5}}{2} \approx -5.646 \).
- \( \sqrt{3} \approx 1.732 \) gives \( -4 - 2\sqrt{3} \approx -7.464 \) and \( -4 + 2\sqrt{3} \approx -0.536 \).

2. **Intersection**:
The intersection of the intervals is:
\[ \left( -7.464, -5.646 \right) \]
The integral values in this range are \( -7, -6 \).

### Step 5: Calculate the sum of integral values

The sum of the integral values \( -7 \) and \( -6 \) is:
\[ -7 + (-6) = -13 \]

### Final Answer
The sum of all the integral values of \( a \) must be equal to \( -13 \).