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

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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
The solutions are $$ p = -7 + \sqrt{87} $$ and $$ p = -7 - \sqrt{87} $$.

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

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