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 = 1.5
1 + -2p + p2 = 1.5

Factor a perfect square on the left side:
(p + -1)(p + -1) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(p + -1) equal to 1.224744871 and -1.224744871.

Subproblem 1
p + -1 = 1.224744871

Simplifying
p + -1 = 1.224744871

Reorder the terms:
-1 + p = 1.224744871

Solving
-1 + p = 1.224744871

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 = 1.224744871 + 1

Combine like terms: -1 + 1 = 0
0 + p = 1.224744871 + 1
p = 1.224744871 + 1

Combine like terms: 1.224744871 + 1 = 2.224744871
p = 2.224744871

Simplifying
p = 2.224744871

Subproblem 2
p + -1 = -1.224744871

Simplifying
p + -1 = -1.224744871

Reorder the terms:
-1 + p = -1.224744871

Solving
-1 + p = -1.224744871

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 = -1.224744871 + 1

Combine like terms: -1 + 1 = 0
0 + p = -1.224744871 + 1
p = -1.224744871 + 1

Combine like terms: -1.224744871 + 1 = -0.224744871
p = -0.224744871

Simplifying
p = -0.224744871

Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {2.224744871, -0.224744871}