3z+15z^2-19z^2+6+11-18z-5z=0

Solution for 3z+15z^2-19z^2+6+11-18z-5z=0 equation:

Simplifying
3z + 15z2 + -19z2 + 6 + 11 + -18z + -5z = 0

Reorder the terms:
6 + 11 + 3z + -18z + -5z + 15z2 + -19z2 = 0

Combine like terms: 6 + 11 = 17
17 + 3z + -18z + -5z + 15z2 + -19z2 = 0

Combine like terms: 3z + -18z = -15z
17 + -15z + -5z + 15z2 + -19z2 = 0

Combine like terms: -15z + -5z = -20z
17 + -20z + 15z2 + -19z2 = 0

Combine like terms: 15z2 + -19z2 = -4z2
17 + -20z + -4z2 = 0

Solving
17 + -20z + -4z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-4.25 + 5z + z2 = 0

Move the constant term to the right:

Add '4.25' to each side of the equation.
-4.25 + 5z + 4.25 + z2 = 0 + 4.25

Reorder the terms:
-4.25 + 4.25 + 5z + z2 = 0 + 4.25

Combine like terms: -4.25 + 4.25 = 0.00
0.00 + 5z + z2 = 0 + 4.25
5z + z2 = 0 + 4.25

Combine like terms: 0 + 4.25 = 4.25
5z + z2 = 4.25

The z term is 5z.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5z + 6.25 + z2 = 4.25 + 6.25

Reorder the terms:
6.25 + 5z + z2 = 4.25 + 6.25

Combine like terms: 4.25 + 6.25 = 10.5
6.25 + 5z + z2 = 10.5

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 10.5

Calculate the square root of the right side: 3.240370349

Break this problem into two subproblems by setting 
(z + 2.5) equal to 3.240370349 and -3.240370349.

Subproblem 1
z + 2.5 = 3.240370349

Simplifying
z + 2.5 = 3.240370349

Reorder the terms:
2.5 + z = 3.240370349

Solving
2.5 + z = 3.240370349

Solving for variable 'z'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + z = 3.240370349 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + z = 3.240370349 + -2.5
z = 3.240370349 + -2.5

Combine like terms: 3.240370349 + -2.5 = 0.740370349
z = 0.740370349

Simplifying
z = 0.740370349

Subproblem 2
z + 2.5 = -3.240370349

Simplifying
z + 2.5 = -3.240370349

Reorder the terms:
2.5 + z = -3.240370349

Solving
2.5 + z = -3.240370349

Solving for variable 'z'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + z = -3.240370349 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + z = -3.240370349 + -2.5
z = -3.240370349 + -2.5

Combine like terms: -3.240370349 + -2.5 = -5.740370349
z = -5.740370349

Simplifying
z = -5.740370349

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.740370349, -5.740370349}