2p^2+4p+1=0

Solution for 2p^2+4p+1=0 equation:

Simplifying
2p2 + 4p + 1 = 0

Reorder the terms:
1 + 4p + 2p2 = 0

Solving
1 + 4p + 2p2 = 0

Solving for variable 'p'.

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
0.5 + 2p + p2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2p + -0.5 + p2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2p + p2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2p + p2 = 0 + -0.5
2p + p2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2p + p2 = -0.5

The p term is 2p.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2p + 1 + p2 = -0.5 + 1

Reorder the terms:
1 + 2p + p2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + 2p + p2 = 0.5

Factor a perfect square on the left side:
(p + 1)(p + 1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(p + 1) equal to 0.707106781 and -0.707106781.

Subproblem 1
p + 1 = 0.707106781

Simplifying
p + 1 = 0.707106781

Reorder the terms:
1 + p = 0.707106781

Solving
1 + p = 0.707106781

Solving for variable 'p'.

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

Add '-1' to each side of the equation.
1 + -1 + p = 0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + p = 0.707106781 + -1
p = 0.707106781 + -1

Combine like terms: 0.707106781 + -1 = -0.292893219
p = -0.292893219

Simplifying
p = -0.292893219

Subproblem 2
p + 1 = -0.707106781

Simplifying
p + 1 = -0.707106781

Reorder the terms:
1 + p = -0.707106781

Solving
1 + p = -0.707106781

Solving for variable 'p'.

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

Add '-1' to each side of the equation.
1 + -1 + p = -0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + p = -0.707106781 + -1
p = -0.707106781 + -1

Combine like terms: -0.707106781 + -1 = -1.707106781
p = -1.707106781

Simplifying
p = -1.707106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {-0.292893219, -1.707106781}