0=p^2+6p+1

Solution for 0=p^2+6p+1 equation:

Simplifying
0 = p2 + 6p + 1

Reorder the terms:
0 = 1 + 6p + p2

Solving
0 = 1 + 6p + p2

Solving for variable 'p'.

Combine like terms: 0 + -1 = -1
-1 + -6p + -1p2 = 1 + 6p + p2 + -1 + -6p + -1p2

Reorder the terms:
-1 + -6p + -1p2 = 1 + -1 + 6p + -6p + p2 + -1p2

Combine like terms: 1 + -1 = 0
-1 + -6p + -1p2 = 0 + 6p + -6p + p2 + -1p2
-1 + -6p + -1p2 = 6p + -6p + p2 + -1p2

Combine like terms: 6p + -6p = 0
-1 + -6p + -1p2 = 0 + p2 + -1p2
-1 + -6p + -1p2 = p2 + -1p2

Combine like terms: p2 + -1p2 = 0
-1 + -6p + -1p2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 6p + p2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(1 + 6p + p2)' equal to zero and attempt to solve:

Simplifying
1 + 6p + p2 = 0

Solving
1 + 6p + p2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 6p + -1 + p2 = 0 + -1

Reorder the terms:
1 + -1 + 6p + p2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 6p + p2 = 0 + -1
6p + p2 = 0 + -1

Combine like terms: 0 + -1 = -1
6p + p2 = -1

The p term is 6p.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6p + 9 + p2 = -1 + 9

Reorder the terms:
9 + 6p + p2 = -1 + 9

Combine like terms: -1 + 9 = 8
9 + 6p + p2 = 8

Factor a perfect square on the left side:
(p + 3)(p + 3) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(p + 3) equal to 2.828427125 and -2.828427125.

Subproblem 1
p + 3 = 2.828427125

Simplifying
p + 3 = 2.828427125

Reorder the terms:
3 + p = 2.828427125

Solving
3 + p = 2.828427125

Solving for variable 'p'.

Move all terms containing p to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + p = 2.828427125 + -3

Combine like terms: 3 + -3 = 0
0 + p = 2.828427125 + -3
p = 2.828427125 + -3

Combine like terms: 2.828427125 + -3 = -0.171572875
p = -0.171572875

Simplifying
p = -0.171572875

Subproblem 2
p + 3 = -2.828427125

Simplifying
p + 3 = -2.828427125

Reorder the terms:
3 + p = -2.828427125

Solving
3 + p = -2.828427125

Solving for variable 'p'.

Move all terms containing p to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + p = -2.828427125 + -3

Combine like terms: 3 + -3 = 0
0 + p = -2.828427125 + -3
p = -2.828427125 + -3

Combine like terms: -2.828427125 + -3 = -5.828427125
p = -5.828427125

Simplifying
p = -5.828427125

Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {-0.171572875, -5.828427125}
Solution
p = {-0.171572875, -5.828427125}