If `a in [-20,0]`, then the probability that the graph of the function `y=16x^(2)+8(a+5)x-7a-5` touches or above the x-aixs is

Options

A. `(3)/(20)`
B. `(13)/(20)`
C. `(7)/(20)`
D. `(1)/(2)`

Explanation

To solve the problem, we need to determine the conditions under which the graph of the function \( y = 16x^2 + 8(a + 5)x - 7a - 5 \) touches or is above the x-axis. This occurs when the discriminant of the quadratic equation is less than or equal to zero.

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation**:
The given function can be expressed in the standard form \( y = Ax^2 + Bx + C \), where:
- \( A = 16 \)
- \( B = 8(a + 5) \)
- \( C = -7a - 5 \)

2. **Set up the discriminant condition**:
For the quadratic to touch or be above the x-axis, the discriminant \( D \) must satisfy:
\[ D = B^2 - 4AC \leq 0 \]

3. **Calculate the discriminant**:
Substitute \( A \), \( B \), and \( C \) into the discriminant formula:
\[ D = [8(a + 5)]^2 - 4 \cdot 16 \cdot (-7a - 5) \]
Simplifying this:
\[ D = 64(a + 5)^2 + 64(7a + 5) \]
\[ D = 64[(a + 5)^2 + (7a + 5)] \]

4. **Combine and simplify the expression**:
Expand \( (a + 5)^2 \):
\[ (a + 5)^2 = a^2 + 10a + 25 \]
Therefore:
\[ D = 64[a^2 + 10a + 25 + 7a + 5] \]
\[ D = 64[a^2 + 17a + 30] \]

5. **Set the discriminant less than or equal to zero**:
We need:
\[ a^2 + 17a + 30 \leq 0 \]

6. **Factor the quadratic**:
The quadratic can be factored as:
\[ (a + 15)(a + 2) \leq 0 \]

7. **Find the critical points**:
The critical points are \( a = -15 \) and \( a = -2 \).

8. **Determine the intervals**:
We analyze the sign of the expression \( (a + 15)(a + 2) \):
- For \( a < -15 \): both factors are negative, product is positive.
- For \( -15 < a < -2 \): one factor is positive, one is negative, product is negative.
- For \( a > -2 \): both factors are positive, product is positive.

Therefore, the inequality \( (a + 15)(a + 2) \leq 0 \) holds for:
\[ a \in [-15, -2] \]

9. **Intersect with the given range**:
We are given \( a \in [-20, 0] \). The intersection of \( [-15, -2] \) and \( [-20, 0] \) is:
\[ a \in [-15, -2] \]

10. **Calculate the length of the interval**:
The length of the interval \( [-15, -2] \) is:
\[ -2 - (-15) = 13 \]

11. **Calculate the total length of the sample space**:
The total length of the interval \( [-20, 0] \) is:
\[ 0 - (-20) = 20 \]

12. **Calculate the probability**:
The probability that the graph touches or is above the x-axis is given by:
\[ P = \frac{\text{Length of favorable interval}}{\text{Total length of sample space}} = \frac{13}{20} \]

### Final Answer:
The probability that the graph of the function touches or is above the x-axis is \( \frac{13}{20} \).