(z^2)+3z+1=0

Solution for (z^2)+3z+1=0 equation:

Simplifying
(z2) + 3z + 1 = 0
z2 + 3z + 1 = 0

Reorder the terms:
1 + 3z + z2 = 0

Solving
1 + 3z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 3z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 3z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 3z + z2 = 0 + -1
3z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
3z + z2 = -1

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 = -1 + 2.25

Reorder the terms:
2.25 + 3z + z2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3z + z2 = 1.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
z + 1.5 = 1.118033989

Simplifying
z + 1.5 = 1.118033989

Reorder the terms:
1.5 + z = 1.118033989

Solving
1.5 + z = 1.118033989

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 = 1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = 1.118033989 + -1.5
z = 1.118033989 + -1.5

Combine like terms: 1.118033989 + -1.5 = -0.381966011
z = -0.381966011

Simplifying
z = -0.381966011

Subproblem 2
z + 1.5 = -1.118033989

Simplifying
z + 1.5 = -1.118033989

Reorder the terms:
1.5 + z = -1.118033989

Solving
1.5 + z = -1.118033989

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 = -1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + z = -1.118033989 + -1.5
z = -1.118033989 + -1.5

Combine like terms: -1.118033989 + -1.5 = -2.618033989
z = -2.618033989

Simplifying
z = -2.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.381966011, -2.618033989}