36z^2+96z+40=0

Solution for 36z^2+96z+40=0 equation:

Simplifying
36z2 + 96z + 40 = 0

Reorder the terms:
40 + 96z + 36z2 = 0

Solving
40 + 96z + 36z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '4'.
4(10 + 24z + 9z2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(10 + 24z + 9z2)' equal to zero and attempt to solve:

Simplifying
10 + 24z + 9z2 = 0

Solving
10 + 24z + 9z2 = 0

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

Divide each side by '9'.
1.111111111 + 2.666666667z + z2 = 0

Move the constant term to the right:

Add '-1.111111111' to each side of the equation.
1.111111111 + 2.666666667z + -1.111111111 + z2 = 0 + -1.111111111

Reorder the terms:
1.111111111 + -1.111111111 + 2.666666667z + z2 = 0 + -1.111111111

Combine like terms: 1.111111111 + -1.111111111 = 0.000000000
0.000000000 + 2.666666667z + z2 = 0 + -1.111111111
2.666666667z + z2 = 0 + -1.111111111

Combine like terms: 0 + -1.111111111 = -1.111111111
2.666666667z + z2 = -1.111111111

The z term is 2.666666667z.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667z + 1.777777780 + z2 = -1.111111111 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667z + z2 = -1.111111111 + 1.777777780

Combine like terms: -1.111111111 + 1.777777780 = 0.666666669
1.777777780 + 2.666666667z + z2 = 0.666666669

Factor a perfect square on the left side:
(z + 1.333333334)(z + 1.333333334) = 0.666666669

Calculate the square root of the right side: 0.816496582

Break this problem into two subproblems by setting 
(z + 1.333333334) equal to 0.816496582 and -0.816496582.

Subproblem 1
z + 1.333333334 = 0.816496582

Simplifying
z + 1.333333334 = 0.816496582

Reorder the terms:
1.333333334 + z = 0.816496582

Solving
1.333333334 + z = 0.816496582

Solving for variable 'z'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + z = 0.816496582 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + z = 0.816496582 + -1.333333334
z = 0.816496582 + -1.333333334

Combine like terms: 0.816496582 + -1.333333334 = -0.516836752
z = -0.516836752

Simplifying
z = -0.516836752

Subproblem 2
z + 1.333333334 = -0.816496582

Simplifying
z + 1.333333334 = -0.816496582

Reorder the terms:
1.333333334 + z = -0.816496582

Solving
1.333333334 + z = -0.816496582

Solving for variable 'z'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + z = -0.816496582 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + z = -0.816496582 + -1.333333334
z = -0.816496582 + -1.333333334

Combine like terms: -0.816496582 + -1.333333334 = -2.149829916
z = -2.149829916

Simplifying
z = -2.149829916

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.516836752, -2.149829916}
Solution
z = {-0.516836752, -2.149829916}