z^2-z-27=0

Solution for z^2-z-27=0 equation:

Simplifying
z2 + -1z + -27 = 0

Reorder the terms:
-27 + -1z + z2 = 0

Solving
-27 + -1z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '27' to each side of the equation.
-27 + -1z + 27 + z2 = 0 + 27

Reorder the terms:
-27 + 27 + -1z + z2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + -1z + z2 = 0 + 27
-1z + z2 = 0 + 27

Combine like terms: 0 + 27 = 27
-1z + z2 = 27

The z term is -1z.  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.
-1z + 0.25 + z2 = 27 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 27 + 0.25

Combine like terms: 27 + 0.25 = 27.25
0.25 + -1z + z2 = 27.25

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 27.25

Calculate the square root of the right side: 5.220153254

Break this problem into two subproblems by setting 
(z + -0.5) equal to 5.220153254 and -5.220153254.

Subproblem 1
z + -0.5 = 5.220153254

Simplifying
z + -0.5 = 5.220153254

Reorder the terms:
-0.5 + z = 5.220153254

Solving
-0.5 + z = 5.220153254

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 = 5.220153254 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = 5.220153254 + 0.5
z = 5.220153254 + 0.5

Combine like terms: 5.220153254 + 0.5 = 5.720153254
z = 5.720153254

Simplifying
z = 5.720153254

Subproblem 2
z + -0.5 = -5.220153254

Simplifying
z + -0.5 = -5.220153254

Reorder the terms:
-0.5 + z = -5.220153254

Solving
-0.5 + z = -5.220153254

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 = -5.220153254 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = -5.220153254 + 0.5
z = -5.220153254 + 0.5

Combine like terms: -5.220153254 + 0.5 = -4.720153254
z = -4.720153254

Simplifying
z = -4.720153254

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {5.720153254, -4.720153254}