3z^2-3z-42=0

Solution for 3z^2-3z-42=0 equation:

Simplifying
3z2 + -3z + -42 = 0

Reorder the terms:
-42 + -3z + 3z2 = 0

Solving
-42 + -3z + 3z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-14 + -1z + z2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-14 + -1z + z2)' equal to zero and attempt to solve:

Simplifying
-14 + -1z + z2 = 0

Solving
-14 + -1z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + -1z + 14 + z2 = 0 + 14

Reorder the terms:
-14 + 14 + -1z + z2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + -1z + z2 = 0 + 14
-1z + z2 = 0 + 14

Combine like terms: 0 + 14 = 14
-1z + z2 = 14

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 14 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 14 + 0.25

Combine like terms: 14 + 0.25 = 14.25
0.25 + -1z + z2 = 14.25

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(z + -0.5) equal to 3.774917218 and -3.774917218.

Subproblem 1
z + -0.5 = 3.774917218

Simplifying
z + -0.5 = 3.774917218

Reorder the terms:
-0.5 + z = 3.774917218

Solving
-0.5 + z = 3.774917218

Solving for variable 'z'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = 3.774917218 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = 3.774917218 + 0.5
z = 3.774917218 + 0.5

Combine like terms: 3.774917218 + 0.5 = 4.274917218
z = 4.274917218

Simplifying
z = 4.274917218

Subproblem 2
z + -0.5 = -3.774917218

Simplifying
z + -0.5 = -3.774917218

Reorder the terms:
-0.5 + z = -3.774917218

Solving
-0.5 + z = -3.774917218

Solving for variable 'z'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = -3.774917218 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = -3.774917218 + 0.5
z = -3.774917218 + 0.5

Combine like terms: -3.774917218 + 0.5 = -3.274917218
z = -3.274917218

Simplifying
z = -3.274917218

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.274917218, -3.274917218}
Solution
z = {4.274917218, -3.274917218}