Questions: Solve the equation for all real solutions in simplest form. 3 p^2 - 3 p - 6 = -2

Solution Steps

To solve the quadratic equation \(3p^2 - 3p - 6 = -2\), first move all terms to one side to set the equation to zero. Then, simplify the equation and use the quadratic formula to find the solutions for \(p\).

We start with the equation: \[ 3p^2 - 3p - 6 = -2 \] Rearranging gives us: \[ 3p^2 - 3p - 4 = 0 \]

The quadratic formula is given by: \[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For our equation, \(a = 3\), \(b = -3\), and \(c = -4\). Plugging in these values, we calculate: \[ p = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3} \] This simplifies to: \[ p = \frac{3 \pm \sqrt{9 + 48}}{6} = \frac{3 \pm \sqrt{57}}{6} \]

Thus, the solutions for \(p\) are: \[ p = \frac{1}{2} - \frac{\sqrt{57}}{6} \quad \text{and} \quad p = \frac{1}{2} + \frac{\sqrt{57}}{6} \]

Final Answer