To solve the quadratic equation \(2p^2 - 7p - 13 = 0\) using the quadratic formula, we start by identifying the coefficients from the standard form of a quadratic equation, \(ap^2 + bp + c = 0\):
\[ a = 2, \quad b = -7, \quad c = -13 \]
The quadratic formula is given by:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
We will first calculate the discriminant \(b^2 - 4ac\):
\[ b^2 = (-7)^2 = 49 \] \[ 4ac = 4 \times 2 \times (-13) = -104 \] \[ b^2 - 4ac = 49 - (-104) = 49 + 104 = 153 \]
Now substituting \(b\), \(a\), and the discriminant into the quadratic formula:
\[ p = \frac{-(-7) \pm \sqrt{153}}{2 \times 2} \]
This simplifies to:
\[ p = \frac{7 \pm \sqrt{153}}{4} \]
Thus, the solutions are:
\[ p = \frac{7 + \sqrt{153}}{4} \quad \text{and} \quad p = \frac{7 - \sqrt{153}}{4} \]
Based on the options provided, we can relate these solutions to the options:
- Option: \(p = \frac{7 + \sqrt{153}}{4}, \frac{7 - \sqrt{153}}{4}\)
So the correct answer is:
\[ p = \frac{7 + \sqrt{153}}{4}, \quad p = \frac{7 - \sqrt{153}}{4} \]