The graph of the function `y=16x^(2)+8(a+2)x-3a-2` is strictly above the x - axis, then number of integral velues of 'a' is

Answer

To solve the problem, we need to analyze the quadratic function given by:

\[ y = 16x^2 + 8(a + 2)x - (3a + 2) \]

We want to find the integral values of \( a \) such that the graph of this function is strictly above the x-axis. This means that the quadratic must not have any real roots, which occurs when the discriminant \( D \) is less than zero.

### Step 1: Identify the coefficients
In the quadratic equation \( ax^2 + bx + c \), we have:
- \( a = 16 \)
- \( b = 8(a + 2) \)
- \( c = -(3a + 2) \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = [8(a + 2)]^2 - 4 \cdot 16 \cdot (-(3a + 2)) \]

### Step 3: Simplify the discriminant
Calculating \( D \):

\[ D = 64(a + 2)^2 + 64(3a + 2) \]

Factoring out 64:

\[ D = 64 \left[ (a + 2)^2 + (3a + 2) \right] \]

Expanding the terms inside the brackets:

\[ D = 64 \left[ a^2 + 4a + 4 + 3a + 2 \right] \]

Combining like terms:

\[ D = 64 \left[ a^2 + 7a + 6 \right] \]

### Step 4: Set the discriminant less than zero
For the quadratic to be strictly above the x-axis, we need:

\[ 64(a^2 + 7a + 6) < 0 \]

This simplifies to:

\[ a^2 + 7a + 6 < 0 \]

### Step 5: Factor the quadratic
Factoring \( a^2 + 7a + 6 \):

\[ a^2 + 7a + 6 = (a + 1)(a + 6) \]

### Step 6: Find the roots and test intervals
The roots of the equation are \( a = -1 \) and \( a = -6 \). We need to find the intervals where the product is negative:

1. \( a < -6 \) (both factors negative, product positive)
2. \( -6 < a < -1 \) (one factor negative, one positive, product negative)
3. \( a > -1 \) (both factors positive, product positive)

Thus, the quadratic \( (a + 1)(a + 6) < 0 \) holds true for:

\[ -6 < a < -1 \]

### Step 7: Identify integral values of \( a \)
The integral values of \( a \) in the interval \( -6 < a < -1 \) are:

- \( -5 \)
- \( -4 \)
- \( -3 \)
- \( -2 \)

### Conclusion
The number of integral values of \( a \) that satisfy the condition is **4**.

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