4z^2+1=8z

Solution for 4z^2+1=8z equation:

Simplifying
4z2 + 1 = 8z

Reorder the terms:
1 + 4z2 = 8z

Solving
1 + 4z2 = 8z

Solving for variable 'z'.

Reorder the terms:
1 + -8z + 4z2 = 8z + -8z

Combine like terms: 8z + -8z = 0
1 + -8z + 4z2 = 0

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

Divide each side by '4'.
0.25 + -2z + z2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -2z + -0.25 + z2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -2z + z2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -2z + z2 = 0 + -0.25
-2z + z2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-2z + z2 = -0.25

The z term is -2z.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2z + 1 + z2 = -0.25 + 1

Reorder the terms:
1 + -2z + z2 = -0.25 + 1

Combine like terms: -0.25 + 1 = 0.75
1 + -2z + z2 = 0.75

Factor a perfect square on the left side:
(z + -1)(z + -1) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(z + -1) equal to 0.866025404 and -0.866025404.

Subproblem 1
z + -1 = 0.866025404

Simplifying
z + -1 = 0.866025404

Reorder the terms:
-1 + z = 0.866025404

Solving
-1 + z = 0.866025404

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = 0.866025404 + 1

Combine like terms: -1 + 1 = 0
0 + z = 0.866025404 + 1
z = 0.866025404 + 1

Combine like terms: 0.866025404 + 1 = 1.866025404
z = 1.866025404

Simplifying
z = 1.866025404

Subproblem 2
z + -1 = -0.866025404

Simplifying
z + -1 = -0.866025404

Reorder the terms:
-1 + z = -0.866025404

Solving
-1 + z = -0.866025404

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = -0.866025404 + 1

Combine like terms: -1 + 1 = 0
0 + z = -0.866025404 + 1
z = -0.866025404 + 1

Combine like terms: -0.866025404 + 1 = 0.133974596
z = 0.133974596

Simplifying
z = 0.133974596

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.866025404, 0.133974596}