Solve each equation by completing the square. 1) \( p^{2}+14 p-38=0 \)

To solve the quadratic equation \( p^{2} + 14p - 38 = 0 \) by completing the square, follow these steps: 1. **Move the constant term to the other side:** \[ p^{2} + 14p = 38 \] 2. **Complete the square:** - Take the coefficient of \( p \), which is 14. - Divide it by 2: \( \frac{14}{2} = 7 \). - Square it: \( 7^{2} = 49 \). Add 49 to both sides to complete the square: \[ p^{2} + 14p + 49 = 38 + 49 \] \[ (p + 7)^2 = 87 \] 3. **Solve for \( p \):** - Take the square root of both sides: \[ p + 7 = \pm \sqrt{87} \] - Subtract 7 from both sides: \[ p = -7 \pm \sqrt{87} \] 4. **Simplify the square root (if possible):** - \( \sqrt{87} \) cannot be simplified further since 87 factors into \( 3 \times 29 \), neither of which is a perfect square. **Final Solutions:** \[ p = -7 + \sqrt{87} \quad \text{and} \quad p = -7 - \sqrt{87} \] Or succinctly written as: \[ p = -7 \pm \sqrt{87} \] Answer: After simplifying, the solutions are –7 plus and minus the square root of 87. Thus, p = –7 ± √87