1+2z-z^2=0

Solution for 1+2z-z^2=0 equation:

Simplifying
1 + 2z + -1z2 = 0

Solving
1 + 2z + -1z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + -2z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + -2z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2z + z2 = 0 + 1
-2z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2z + z2 = 1

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

Add '1' to each side of the equation.
-2z + 1 + z2 = 1 + 1

Reorder the terms:
1 + -2z + z2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + -2z + z2 = 2

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

Calculate the square root of the right side: 1.414213562

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

Subproblem 1
z + -1 = 1.414213562

Simplifying
z + -1 = 1.414213562

Reorder the terms:
-1 + z = 1.414213562

Solving
-1 + z = 1.414213562

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = 1.414213562 + 1

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

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

Simplifying
z = 2.414213562

Subproblem 2
z + -1 = -1.414213562

Simplifying
z + -1 = -1.414213562

Reorder the terms:
-1 + z = -1.414213562

Solving
-1 + z = -1.414213562

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = -1.414213562 + 1

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

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

Simplifying
z = -0.414213562

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