4z^2-2z-5=0

Solution for 4z^2-2z-5=0 equation:

Simplifying
4z2 + -2z + -5 = 0

Reorder the terms:
-5 + -2z + 4z2 = 0

Solving
-5 + -2z + 4z2 = 0

Solving for variable 'z'.

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

Divide each side by '4'.
-1.25 + -0.5z + z2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -0.5z + 1.25 + z2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -0.5z + z2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -0.5z + z2 = 0 + 1.25
-0.5z + z2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-0.5z + z2 = 1.25

The z term is -0.5z.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5z + 0.0625 + z2 = 1.25 + 0.0625

Reorder the terms:
0.0625 + -0.5z + z2 = 1.25 + 0.0625

Combine like terms: 1.25 + 0.0625 = 1.3125
0.0625 + -0.5z + z2 = 1.3125

Factor a perfect square on the left side:
(z + -0.25)(z + -0.25) = 1.3125

Calculate the square root of the right side: 1.145643924

Break this problem into two subproblems by setting 
(z + -0.25) equal to 1.145643924 and -1.145643924.

Subproblem 1
z + -0.25 = 1.145643924

Simplifying
z + -0.25 = 1.145643924

Reorder the terms:
-0.25 + z = 1.145643924

Solving
-0.25 + z = 1.145643924

Solving for variable 'z'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + z = 1.145643924 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + z = 1.145643924 + 0.25
z = 1.145643924 + 0.25

Combine like terms: 1.145643924 + 0.25 = 1.395643924
z = 1.395643924

Simplifying
z = 1.395643924

Subproblem 2
z + -0.25 = -1.145643924

Simplifying
z + -0.25 = -1.145643924

Reorder the terms:
-0.25 + z = -1.145643924

Solving
-0.25 + z = -1.145643924

Solving for variable 'z'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + z = -1.145643924 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + z = -1.145643924 + 0.25
z = -1.145643924 + 0.25

Combine like terms: -1.145643924 + 0.25 = -0.895643924
z = -0.895643924

Simplifying
z = -0.895643924

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.395643924, -0.895643924}