z^2-4z-27=0

Solution for z^2-4z-27=0 equation:

Simplifying
z2 + -4z + -27 = 0

Reorder the terms:
-27 + -4z + z2 = 0

Solving
-27 + -4z + 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 + -4z + 27 + z2 = 0 + 27

Reorder the terms:
-27 + 27 + -4z + z2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + -4z + z2 = 0 + 27
-4z + z2 = 0 + 27

Combine like terms: 0 + 27 = 27
-4z + z2 = 27

The z term is -4z.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4z + 4 + z2 = 27 + 4

Reorder the terms:
4 + -4z + z2 = 27 + 4

Combine like terms: 27 + 4 = 31
4 + -4z + z2 = 31

Factor a perfect square on the left side:
(z + -2)(z + -2) = 31

Calculate the square root of the right side: 5.567764363

Break this problem into two subproblems by setting 
(z + -2) equal to 5.567764363 and -5.567764363.

Subproblem 1
z + -2 = 5.567764363

Simplifying
z + -2 = 5.567764363

Reorder the terms:
-2 + z = 5.567764363

Solving
-2 + z = 5.567764363

Solving for variable 'z'.

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

Add '2' to each side of the equation.
-2 + 2 + z = 5.567764363 + 2

Combine like terms: -2 + 2 = 0
0 + z = 5.567764363 + 2
z = 5.567764363 + 2

Combine like terms: 5.567764363 + 2 = 7.567764363
z = 7.567764363

Simplifying
z = 7.567764363

Subproblem 2
z + -2 = -5.567764363

Simplifying
z + -2 = -5.567764363

Reorder the terms:
-2 + z = -5.567764363

Solving
-2 + z = -5.567764363

Solving for variable 'z'.

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

Add '2' to each side of the equation.
-2 + 2 + z = -5.567764363 + 2

Combine like terms: -2 + 2 = 0
0 + z = -5.567764363 + 2
z = -5.567764363 + 2

Combine like terms: -5.567764363 + 2 = -3.567764363
z = -3.567764363

Simplifying
z = -3.567764363

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {7.567764363, -3.567764363}