-8z^2+2z+4=0

Solution for -8z^2+2z+4=0 equation:

Simplifying
-8z2 + 2z + 4 = 0

Reorder the terms:
4 + 2z + -8z2 = 0

Solving
4 + 2z + -8z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '2'.
2(2 + z + -4z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(2 + z + -4z2)' equal to zero and attempt to solve:

Simplifying
2 + z + -4z2 = 0

Solving
2 + z + -4z2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-0.5 + -0.25z + z2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.25z + 0.5 + z2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.25z + z2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.25z + z2 = 0 + 0.5
-0.25z + z2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.25z + z2 = 0.5

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.
-0.25z + 0.25 + z2 = 0.5 + 0.25

Reorder the terms:
0.25 + -0.25z + z2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
0.25 + -0.25z + z2 = 0.75

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(z + 0.5) equal to 0.866025404 and -0.866025404.

Subproblem 1
z + 0.5 = 0.866025404

Simplifying
z + 0.5 = 0.866025404

Reorder the terms:
0.5 + z = 0.866025404

Solving
0.5 + z = 0.866025404

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 = 0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 0.866025404 + -0.5
z = 0.866025404 + -0.5

Combine like terms: 0.866025404 + -0.5 = 0.366025404
z = 0.366025404

Simplifying
z = 0.366025404

Subproblem 2
z + 0.5 = -0.866025404

Simplifying
z + 0.5 = -0.866025404

Reorder the terms:
0.5 + z = -0.866025404

Solving
0.5 + z = -0.866025404

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 = -0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -0.866025404 + -0.5
z = -0.866025404 + -0.5

Combine like terms: -0.866025404 + -0.5 = -1.366025404
z = -1.366025404

Simplifying
z = -1.366025404

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.366025404, -1.366025404}
Solution
z = {0.366025404, -1.366025404}