Find the roots of each equation. 2p^2+7p-30=0

To find the roots of the quadratic equation \(2p^2 + 7p - 30 = 0\), we can use the quadratic formula:<br /><br />\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br /><br />Here, \(a = 2\), \(b = 7\), and \(c = -30\).<br /><br />First, we calculate the discriminant (\(\Delta\)):<br /><br />\[ \Delta = b^2 - 4ac \]<br />\[ \Delta = 7^2 - 4 \cdot 2 \cdot (-30) \]<br />\[ \Delta = 49 + 240 \]<br />\[ \Delta = 289 \]<br /><br />Now, we can find the roots using the quadratic formula:<br /><br />\[ p = \frac{-b \pm \sqrt{\Delta}}{2a} \]<br />\[ p = \frac{-7 \pm \sqrt{289}}{4} \]<br />\[ p = \frac{-7 \pm 17}{4} \]<br /><br />This gives us two solutions:<br /><br />1. \( p = \frac{-7 + 17}{4} = \frac{10}{4} = \frac{5}{2} \)<br />2. \( p = \frac{-7 - 17}{4} = \frac{-24}{4} = -6 \)<br /><br />So, the roots of the equation \(2p^2 + 7p - 30 = 0\) are:<br /><br />\[ p = \frac{5}{2} \quad \text{and} \quad p = -6 \]