2z^2+9z+3=0

Solution for 2z^2+9z+3=0 equation:

Simplifying
2z2 + 9z + 3 = 0

Reorder the terms:
3 + 9z + 2z2 = 0

Solving
3 + 9z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
1.5 + 4.5z + z2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 4.5z + -1.5 + z2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 4.5z + z2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 4.5z + z2 = 0 + -1.5
4.5z + z2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
4.5z + z2 = -1.5

The z term is 4.5z.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5z + 5.0625 + z2 = -1.5 + 5.0625

Reorder the terms:
5.0625 + 4.5z + z2 = -1.5 + 5.0625

Combine like terms: -1.5 + 5.0625 = 3.5625
5.0625 + 4.5z + z2 = 3.5625

Factor a perfect square on the left side:
(z + 2.25)(z + 2.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(z + 2.25) equal to 1.887458609 and -1.887458609.

Subproblem 1
z + 2.25 = 1.887458609

Simplifying
z + 2.25 = 1.887458609

Reorder the terms:
2.25 + z = 1.887458609

Solving
2.25 + z = 1.887458609

Solving for variable 'z'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + z = 1.887458609 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + z = 1.887458609 + -2.25
z = 1.887458609 + -2.25

Combine like terms: 1.887458609 + -2.25 = -0.362541391
z = -0.362541391

Simplifying
z = -0.362541391

Subproblem 2
z + 2.25 = -1.887458609

Simplifying
z + 2.25 = -1.887458609

Reorder the terms:
2.25 + z = -1.887458609

Solving
2.25 + z = -1.887458609

Solving for variable 'z'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + z = -1.887458609 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + z = -1.887458609 + -2.25
z = -1.887458609 + -2.25

Combine like terms: -1.887458609 + -2.25 = -4.137458609
z = -4.137458609

Simplifying
z = -4.137458609

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.362541391, -4.137458609}