z^2-3z-6=0

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

Simplifying
z2 + -3z + -6 = 0

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

Solving
-6 + -3z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -3z + 6 + z2 = 0 + 6

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

Combine like terms: -6 + 6 = 0
0 + -3z + z2 = 0 + 6
-3z + z2 = 0 + 6

Combine like terms: 0 + 6 = 6
-3z + z2 = 6

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 = 6 + 2.25

Reorder the terms:
2.25 + -3z + z2 = 6 + 2.25

Combine like terms: 6 + 2.25 = 8.25
2.25 + -3z + z2 = 8.25

Factor a perfect square on the left side:
(z + -1.5)(z + -1.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(z + -1.5) equal to 2.872281323 and -2.872281323.

Subproblem 1
z + -1.5 = 2.872281323

Simplifying
z + -1.5 = 2.872281323

Reorder the terms:
-1.5 + z = 2.872281323

Solving
-1.5 + z = 2.872281323

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 = 2.872281323 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + z = 2.872281323 + 1.5
z = 2.872281323 + 1.5

Combine like terms: 2.872281323 + 1.5 = 4.372281323
z = 4.372281323

Simplifying
z = 4.372281323

Subproblem 2
z + -1.5 = -2.872281323

Simplifying
z + -1.5 = -2.872281323

Reorder the terms:
-1.5 + z = -2.872281323

Solving
-1.5 + z = -2.872281323

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 = -2.872281323 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + z = -2.872281323 + 1.5
z = -2.872281323 + 1.5

Combine like terms: -2.872281323 + 1.5 = -1.372281323
z = -1.372281323

Simplifying
z = -1.372281323

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.372281323, -1.372281323}