2z^2-6z-3=0

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

Simplifying
2z2 + -6z + -3 = 0

Reorder the terms:
-3 + -6z + 2z2 = 0

Solving
-3 + -6z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
-1.5 + -3z + z2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -3z + 1.5 + z2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -3z + z2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -3z + z2 = 0 + 1.5
-3z + z2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-3z + z2 = 1.5

The z term is -3z.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3z + 2.25 + z2 = 1.5 + 2.25

Reorder the terms:
2.25 + -3z + z2 = 1.5 + 2.25

Combine like terms: 1.5 + 2.25 = 3.75
2.25 + -3z + z2 = 3.75

Factor a perfect square on the left side:
(z + -1.5)(z + -1.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(z + -1.5) equal to 1.936491673 and -1.936491673.

Subproblem 1
z + -1.5 = 1.936491673

Simplifying
z + -1.5 = 1.936491673

Reorder the terms:
-1.5 + z = 1.936491673

Solving
-1.5 + z = 1.936491673

Solving for variable 'z'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + z = 1.936491673 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + z = 1.936491673 + 1.5
z = 1.936491673 + 1.5

Combine like terms: 1.936491673 + 1.5 = 3.436491673
z = 3.436491673

Simplifying
z = 3.436491673

Subproblem 2
z + -1.5 = -1.936491673

Simplifying
z + -1.5 = -1.936491673

Reorder the terms:
-1.5 + z = -1.936491673

Solving
-1.5 + z = -1.936491673

Solving for variable 'z'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + z = -1.936491673 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + z = -1.936491673 + 1.5
z = -1.936491673 + 1.5

Combine like terms: -1.936491673 + 1.5 = -0.436491673
z = -0.436491673

Simplifying
z = -0.436491673

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.436491673, -0.436491673}