The least value of `n in N` for which `(n - 4)x^(2) + 8x + n + 2 gt 0 AA x in R`, is

Options

A. 11
B. 10
C. 8
D. 7

Answer

To solve the problem, we need to find the least value of \( n \in \mathbb{N} \) such that the quadratic expression \( (n - 4)x^2 + 8x + (n + 2) > 0 \) for all \( x \in \mathbb{R} \).

### Step-by-Step Solution:

1. **Identify the Quadratic Coefficients**:
The given quadratic expression can be rewritten as:
\[ a = n - 4, \quad b = 8, \quad c = n + 2 \]

2. **Condition for the Quadratic to be Always Positive**:
For the quadratic \( ax^2 + bx + c \) to be positive for all \( x \), the following conditions must be satisfied:
- \( a > 0 \)
- The discriminant \( D < 0 \)

3. **Condition 1: \( a > 0 \)**:
\[ n - 4 > 0 \implies n > 4 \]

4. **Condition 2: Discriminant \( D < 0 \)**:
The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = 8^2 - 4(n - 4)(n + 2) \]
Simplifying this:
\[ D = 64 - 4[(n - 4)(n + 2)] \]
Expanding the product:
\[ D = 64 - 4(n^2 + 2n - 4n - 8) = 64 - 4(n^2 - 2n - 8) \]
\[ = 64 - 4n^2 + 8n + 32 = 96 - 4n^2 + 8n \]
Setting the discriminant less than zero:
\[ 96 - 4n^2 + 8n < 0 \]
Rearranging gives:
\[ 4n^2 - 8n - 96 > 0 \]
Dividing through by 4:
\[ n^2 - 2n - 24 > 0 \]

5. **Factoring the Quadratic**:
To factor \( n^2 - 2n - 24 \):
\[ (n - 6)(n + 4) > 0 \]

6. **Finding the Critical Points**:
The critical points are \( n = 6 \) and \( n = -4 \).

7. **Testing Intervals**:
We test the intervals determined by the critical points:
- For \( n < -4 \): Choose \( n = -5 \) → Positive
- For \( -4 < n < 6 \): Choose \( n = 0 \) → Negative
- For \( n > 6 \): Choose \( n = 7 \) → Positive

Thus, the solution to the inequality \( (n - 6)(n + 4) > 0 \) is:
\[ n < -4 \quad \text{or} \quad n > 6 \]

8. **Combining Conditions**:
We already have \( n > 4 \) from the first condition. Therefore, we combine:
\[ n > 6 \]

9. **Finding the Least Natural Number**:
The least natural number satisfying \( n > 6 \) is \( n = 7 \).

### Conclusion:
The least value of \( n \in \mathbb{N} \) for which \( (n - 4)x^2 + 8x + (n + 2) > 0 \) for all \( x \in \mathbb{R} \) is:
\[ \boxed{7} \]