p^2=2p+1

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

Simplifying
p2 = 2p + 1

Reorder the terms:
p2 = 1 + 2p

Solving
p2 = 1 + 2p

Solving for variable 'p'.

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

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

Combine like terms: 1 + -1 = 0
-1 + -2p + p2 = 0 + 2p + -2p
-1 + -2p + p2 = 2p + -2p

Combine like terms: 2p + -2p = 0
-1 + -2p + p2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2p + 1 + p2 = 0 + 1

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

Combine like terms: -1 + 1 = 0
0 + -2p + p2 = 0 + 1
-2p + p2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2p + p2 = 1

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 = 1 + 1

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

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

Factor a perfect square on the left side:
(p + -1)(p + -1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(p + -1) equal to 1.414213562 and -1.414213562.

Subproblem 1
p + -1 = 1.414213562

Simplifying
p + -1 = 1.414213562

Reorder the terms:
-1 + p = 1.414213562

Solving
-1 + p = 1.414213562

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 = 1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + p = 1.414213562 + 1
p = 1.414213562 + 1

Combine like terms: 1.414213562 + 1 = 2.414213562
p = 2.414213562

Simplifying
p = 2.414213562

Subproblem 2
p + -1 = -1.414213562

Simplifying
p + -1 = -1.414213562

Reorder the terms:
-1 + p = -1.414213562

Solving
-1 + p = -1.414213562

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 = -1.414213562 + 1

Combine like terms: -1 + 1 = 0
0 + p = -1.414213562 + 1
p = -1.414213562 + 1

Combine like terms: -1.414213562 + 1 = -0.414213562
p = -0.414213562

Simplifying
p = -0.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {2.414213562, -0.414213562}