If ` x^(2) - px + 4 gt 0 ` for all real value of x,then which one of the following is correct ?

Options

A. `|p| lt 4`
B. `|p| le 4`
C. ` |p| gt4`
D. `|p| ge 4`

Explanation

To solve the problem, we need to determine the conditions under which the quadratic expression \( x^2 - px + 4 \) is greater than zero for all real values of \( x \).

### Step-by-step Solution:

1. **Identify the Quadratic Expression**:
The given expression is \( x^2 - px + 4 \).

2. **Condition for Positivity**:
A quadratic expression \( ax^2 + bx + c \) is positive for all real \( x \) if its discriminant is less than zero. The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = -p \), and \( c = 4 \).

3. **Calculate the Discriminant**:
Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (-p)^2 - 4 \cdot 1 \cdot 4 \]
Simplifying this gives:
\[ D = p^2 - 16 \]

4. **Set the Discriminant Less Than Zero**:
For the quadratic to be positive for all \( x \), we need:
\[ p^2 - 16 < 0 \]

5. **Solve the Inequality**:
Rearranging the inequality:
\[ p^2 < 16 \]
Taking the square root of both sides gives:
\[ |p| < 4 \]
This implies:
\[ -4 < p < 4 \]

6. **Conclusion**:
The condition for \( p \) is that \( p \) must lie within the interval \( (-4, 4) \).

### Final Answer:
Thus, the correct option is that \( p \) must be in the range \( -4 < p < 4 \).