0=z*z-z-5

Solution for 0=z*z-z-5 equation:

Simplifying
0 = z * z + -1z + -5

Multiply z * z
0 = z2 + -1z + -5

Reorder the terms:
0 = -5 + -1z + z2

Solving
0 = -5 + -1z + z2

Solving for variable 'z'.

Combine like terms: 0 + 5 = 5
5 + z + -1z2 = -5 + -1z + z2 + 5 + z + -1z2

Reorder the terms:
5 + z + -1z2 = -5 + 5 + -1z + z + z2 + -1z2

Combine like terms: -5 + 5 = 0
5 + z + -1z2 = 0 + -1z + z + z2 + -1z2
5 + z + -1z2 = -1z + z + z2 + -1z2

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

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

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

Divide each side by '-1'.
-5 + -1z + z2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1z + 5 + z2 = 0 + 5

Reorder the terms:
-5 + 5 + -1z + z2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -1z + z2 = 0 + 5
-1z + z2 = 0 + 5

Combine like terms: 0 + 5 = 5
-1z + z2 = 5

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.
-1z + 0.25 + z2 = 5 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + -1z + z2 = 5.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(z + 0.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
z + 0.5 = 2.291287847

Simplifying
z + 0.5 = 2.291287847

Reorder the terms:
0.5 + z = 2.291287847

Solving
0.5 + z = 2.291287847

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 = 2.291287847 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 2.291287847 + -0.5
z = 2.291287847 + -0.5

Combine like terms: 2.291287847 + -0.5 = 1.791287847
z = 1.791287847

Simplifying
z = 1.791287847

Subproblem 2
z + 0.5 = -2.291287847

Simplifying
z + 0.5 = -2.291287847

Reorder the terms:
0.5 + z = -2.291287847

Solving
0.5 + z = -2.291287847

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 = -2.291287847 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -2.291287847 + -0.5
z = -2.291287847 + -0.5

Combine like terms: -2.291287847 + -0.5 = -2.791287847
z = -2.791287847

Simplifying
z = -2.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.791287847, -2.791287847}