3z^2-6z-4=0

Solution for 3z^2-6z-4=0 equation:

Simplifying
3z2 + -6z + -4 = 0

Reorder the terms:
-4 + -6z + 3z2 = 0

Solving
-4 + -6z + 3z2 = 0

Solving for variable 'z'.

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

Divide each side by '3'.
-1.333333333 + -2z + z2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -2z + 1.333333333 + z2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -2z + z2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -2z + z2 = 0 + 1.333333333
-2z + z2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-2z + z2 = 1.333333333

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 = 1.333333333 + 1

Reorder the terms:
1 + -2z + z2 = 1.333333333 + 1

Combine like terms: 1.333333333 + 1 = 2.333333333
1 + -2z + z2 = 2.333333333

Factor a perfect square on the left side:
(z + -1)(z + -1) = 2.333333333

Calculate the square root of the right side: 1.527525232

Break this problem into two subproblems by setting 
(z + -1) equal to 1.527525232 and -1.527525232.

Subproblem 1
z + -1 = 1.527525232

Simplifying
z + -1 = 1.527525232

Reorder the terms:
-1 + z = 1.527525232

Solving
-1 + z = 1.527525232

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 = 1.527525232 + 1

Combine like terms: -1 + 1 = 0
0 + z = 1.527525232 + 1
z = 1.527525232 + 1

Combine like terms: 1.527525232 + 1 = 2.527525232
z = 2.527525232

Simplifying
z = 2.527525232

Subproblem 2
z + -1 = -1.527525232

Simplifying
z + -1 = -1.527525232

Reorder the terms:
-1 + z = -1.527525232

Solving
-1 + z = -1.527525232

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 = -1.527525232 + 1

Combine like terms: -1 + 1 = 0
0 + z = -1.527525232 + 1
z = -1.527525232 + 1

Combine like terms: -1.527525232 + 1 = -0.527525232
z = -0.527525232

Simplifying
z = -0.527525232

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.527525232, -0.527525232}