z^2+z=1

Solution for z^2+z=1 equation:

Simplifying
z2 + z = 1

Reorder the terms:
z + z2 = 1

Solving
z + z2 = 1

Solving for variable 'z'.

Reorder the terms:
-1 + z + z2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + z + z2 = 0 + 1
z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
z + z2 = 1

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
z + 0.25 + z2 = 1 + 0.25

Reorder the terms:
0.25 + z + z2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + z + z2 = 1.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
z + 0.5 = 1.118033989

Simplifying
z + 0.5 = 1.118033989

Reorder the terms:
0.5 + z = 1.118033989

Solving
0.5 + z = 1.118033989

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z = 1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 1.118033989 + -0.5
z = 1.118033989 + -0.5

Combine like terms: 1.118033989 + -0.5 = 0.618033989
z = 0.618033989

Simplifying
z = 0.618033989

Subproblem 2
z + 0.5 = -1.118033989

Simplifying
z + 0.5 = -1.118033989

Reorder the terms:
0.5 + z = -1.118033989

Solving
0.5 + z = -1.118033989

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z = -1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -1.118033989 + -0.5
z = -1.118033989 + -0.5

Combine like terms: -1.118033989 + -0.5 = -1.618033989
z = -1.618033989

Simplifying
z = -1.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.618033989, -1.618033989}