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

Options

A. `|P|lt 4`
B. `|P|le 4`
C. `|P| gt 4`
D. `|P| ge 4`

Answer

To solve the inequality \( x^2 - px + 4 > 0 \) for all real values of \( x \), we need to ensure that the quadratic does not have any real roots. This can be achieved by ensuring that the discriminant of the quadratic equation is less than zero.

### Step-by-Step Solution:

1. **Identify the quadratic equation:**
The given quadratic equation is:
\[ f(x) = x^2 - px + 4 \]

2. **Determine the discriminant:**
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c \) is given by:
\[ D = b^2 - 4ac \]
For our equation, \( a = 1 \), \( b = -p \), and \( c = 4 \). Therefore, the discriminant is:
\[ D = (-p)^2 - 4(1)(4) = p^2 - 16 \]

3. **Set the discriminant less than zero:**
To ensure that the quadratic is positive for all real \( x \), we need:
\[ D < 0 \]
This gives us the inequality:
\[ p^2 - 16 < 0 \]

4. **Solve the inequality:**
Rearranging the inequality:
\[ p^2 < 16 \]
Taking the square root of both sides, we find:
\[ -4 < p < 4 \]

5. **Conclusion:**
The values of \( p \) must lie in the interval:
\[ p \in (-4, 4) \]
Therefore, the correct condition for \( p \) is that \( p \) must be between -4 and 4.