0=35+12z-z^2

Solution for 0=35+12z-z^2 equation:

Simplifying
0 = 35 + 12z + -1z2

Solving
0 = 35 + 12z + -1z2

Solving for variable 'z'.

Combine like terms: 0 + -35 = -35
-35 + -12z + z2 = 35 + 12z + -1z2 + -35 + -12z + z2

Reorder the terms:
-35 + -12z + z2 = 35 + -35 + 12z + -12z + -1z2 + z2

Combine like terms: 35 + -35 = 0
-35 + -12z + z2 = 0 + 12z + -12z + -1z2 + z2
-35 + -12z + z2 = 12z + -12z + -1z2 + z2

Combine like terms: 12z + -12z = 0
-35 + -12z + z2 = 0 + -1z2 + z2
-35 + -12z + z2 = -1z2 + z2

Combine like terms: -1z2 + z2 = 0
-35 + -12z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '35' to each side of the equation.
-35 + -12z + 35 + z2 = 0 + 35

Reorder the terms:
-35 + 35 + -12z + z2 = 0 + 35

Combine like terms: -35 + 35 = 0
0 + -12z + z2 = 0 + 35
-12z + z2 = 0 + 35

Combine like terms: 0 + 35 = 35
-12z + z2 = 35

The z term is -12z.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12z + 36 + z2 = 35 + 36

Reorder the terms:
36 + -12z + z2 = 35 + 36

Combine like terms: 35 + 36 = 71
36 + -12z + z2 = 71

Factor a perfect square on the left side:
(z + -6)(z + -6) = 71

Calculate the square root of the right side: 8.426149773

Break this problem into two subproblems by setting 
(z + -6) equal to 8.426149773 and -8.426149773.

Subproblem 1
z + -6 = 8.426149773

Simplifying
z + -6 = 8.426149773

Reorder the terms:
-6 + z = 8.426149773

Solving
-6 + z = 8.426149773

Solving for variable 'z'.

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

Add '6' to each side of the equation.
-6 + 6 + z = 8.426149773 + 6

Combine like terms: -6 + 6 = 0
0 + z = 8.426149773 + 6
z = 8.426149773 + 6

Combine like terms: 8.426149773 + 6 = 14.426149773
z = 14.426149773

Simplifying
z = 14.426149773

Subproblem 2
z + -6 = -8.426149773

Simplifying
z + -6 = -8.426149773

Reorder the terms:
-6 + z = -8.426149773

Solving
-6 + z = -8.426149773

Solving for variable 'z'.

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

Add '6' to each side of the equation.
-6 + 6 + z = -8.426149773 + 6

Combine like terms: -6 + 6 = 0
0 + z = -8.426149773 + 6
z = -8.426149773 + 6

Combine like terms: -8.426149773 + 6 = -2.426149773
z = -2.426149773

Simplifying
z = -2.426149773

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {14.426149773, -2.426149773}