The probabillity of selecting integers `ain[-5,30]`, such that `x^2+2(a+4)x-5a+64 gt 0` , for all `x in R` is

Answer

To solve the problem, we need to determine the probability of selecting integers \( a \) from the interval \([-5, 30]\) such that the quadratic inequality \( x^2 + 2(a + 4)x - 5a + 64 > 0 \) holds for all \( x \in \mathbb{R} \).

### Step-by-step Solution:

1. **Understanding the Quadratic Inequality**:
The quadratic inequality \( x^2 + 2(a + 4)x - 5a + 64 > 0 \) will be positive for all \( x \) if its discriminant is less than 0.

2. **Finding the Discriminant**:
The standard form of a quadratic equation is \( Ax^2 + Bx + C \), where:
- \( A = 1 \)
- \( B = 2(a + 4) \)
- \( C = -5a + 64 \)

The discriminant \( D \) is given by:
\[ D = B^2 - 4AC \]
Substituting the values:
\[ D = [2(a + 4)]^2 - 4(1)(-5a + 64) \]

3. **Expanding the Discriminant**:
\[ D = 4(a + 4)^2 + 20a - 256 \]
Expanding \( (a + 4)^2 \):
\[ D = 4(a^2 + 8a + 16) + 20a - 256 \]
\[ D = 4a^2 + 32a + 64 + 20a - 256 \]
\[ D = 4a^2 + 52a - 192 \]

4. **Setting the Discriminant Less Than Zero**:
We need:
\[ 4a^2 + 52a - 192 < 0 \]
Dividing the entire inequality by 4:
\[ a^2 + 13a - 48 < 0 \]

5. **Finding the Roots of the Quadratic**:
To find the roots, we use the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( b = 13 \), \( a = 1 \), and \( c = -48 \):
\[ a = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 1 \cdot (-48)}}{2 \cdot 1} \]
\[ a = \frac{-13 \pm \sqrt{169 + 192}}{2} \]
\[ a = \frac{-13 \pm \sqrt{361}}{2} \]
\[ a = \frac{-13 \pm 19}{2} \]
This gives us the roots:
\[ a_1 = 3 \quad \text{and} \quad a_2 = -16 \]

6. **Finding the Interval**:
The quadratic \( a^2 + 13a - 48 \) is negative between its roots:
\[ -16 < a < 3 \]

7. **Considering the Given Range**:
We need to find integers \( a \) in the interval \([-5, 30]\) that also satisfy the interval \((-16, 3)\). The integers in this range are:
\[ -5, -4, -3, -2, -1, 0, 1, 2 \]
This gives us a total of 8 integers.

8. **Calculating the Total Sample Space**:
The total integers in the range \([-5, 30]\) are:
\[ 30 - (-5) + 1 = 36 \]

9. **Calculating the Probability**:
The probability \( P \) is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{8}{36} = \frac{2}{9} \]

### Final Answer:
The probability of selecting integers \( a \) such that the quadratic inequality holds for all \( x \in \mathbb{R} \) is \( \frac{2}{9} \).