6z^2-13z-6=0

Solution for 6z^2-13z-6=0 equation:

Simplifying
6z2 + -13z + -6 = 0

Reorder the terms:
-6 + -13z + 6z2 = 0

Solving
-6 + -13z + 6z2 = 0

Solving for variable 'z'.

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

Divide each side by '6'.
-1 + -2.166666667z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2.166666667z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + -2.166666667z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2.166666667z + z2 = 0 + 1
-2.166666667z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2.166666667z + z2 = 1

The z term is -2.166666667z.  Take half its coefficient (-1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
-2.166666667z + 1.173611113 + z2 = 1 + 1.173611113

Reorder the terms:
1.173611113 + -2.166666667z + z2 = 1 + 1.173611113

Combine like terms: 1 + 1.173611113 = 2.173611113
1.173611113 + -2.166666667z + z2 = 2.173611113

Factor a perfect square on the left side:
(z + -1.083333334)(z + -1.083333334) = 2.173611113

Calculate the square root of the right side: 1.474317168

Break this problem into two subproblems by setting 
(z + -1.083333334) equal to 1.474317168 and -1.474317168.

Subproblem 1
z + -1.083333334 = 1.474317168

Simplifying
z + -1.083333334 = 1.474317168

Reorder the terms:
-1.083333334 + z = 1.474317168

Solving
-1.083333334 + z = 1.474317168

Solving for variable 'z'.

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

Add '1.083333334' to each side of the equation.
-1.083333334 + 1.083333334 + z = 1.474317168 + 1.083333334

Combine like terms: -1.083333334 + 1.083333334 = 0.000000000
0.000000000 + z = 1.474317168 + 1.083333334
z = 1.474317168 + 1.083333334

Combine like terms: 1.474317168 + 1.083333334 = 2.557650502
z = 2.557650502

Simplifying
z = 2.557650502

Subproblem 2
z + -1.083333334 = -1.474317168

Simplifying
z + -1.083333334 = -1.474317168

Reorder the terms:
-1.083333334 + z = -1.474317168

Solving
-1.083333334 + z = -1.474317168

Solving for variable 'z'.

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

Add '1.083333334' to each side of the equation.
-1.083333334 + 1.083333334 + z = -1.474317168 + 1.083333334

Combine like terms: -1.083333334 + 1.083333334 = 0.000000000
0.000000000 + z = -1.474317168 + 1.083333334
z = -1.474317168 + 1.083333334

Combine like terms: -1.474317168 + 1.083333334 = -0.390983834
z = -0.390983834

Simplifying
z = -0.390983834

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.557650502, -0.390983834}