2p² + 7p - 30 =0

To solve the quadratic equation \( 2p^2 + 7p - 30 = 0 \), we can use the quadratic formula:

\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = 2 \), \( b = 7 \), and \( c = -30 \).

1. Calculate the discriminant \( b^2 - 4ac \): \[ b^2 = 7^2 = 49 \] \[ 4ac = 4 \cdot 2 \cdot -30 = -240 \] \[ b^2 - 4ac = 49 - (-240) = 49 + 240 = 289 \]

2. Calculate the roots using the quadratic formula: \[ p = \frac{-7 \pm \sqrt{289}}{2 \cdot 2} \] Since \( \sqrt{289} = 17 \): \[ p = \frac{-7 \pm 17}{4} \]

3. Calculate the two possible values for \( p \):

   • For the positive case: \[ p = \frac{-7 + 17}{4} = \frac{10}{4} = 2.5 \]

   • For the negative case: \[ p = \frac{-7 - 17}{4} = \frac{-24}{4} = -6 \]

Thus, the solutions to the equation \( 2p^2 + 7p - 30 = 0 \) are:

\[ p = 2.5 \quad \text{and} \quad p = -6 \]