3+5z+z^2=0

Solution for 3+5z+z^2=0 equation:

Simplifying
3 + 5z + z2 = 0

Solving
3 + 5z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 5z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + 5z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 5z + z2 = 0 + -3
5z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
5z + z2 = -3

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 = -3 + 6.25

Reorder the terms:
6.25 + 5z + z2 = -3 + 6.25

Combine like terms: -3 + 6.25 = 3.25
6.25 + 5z + z2 = 3.25

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(z + 2.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
z + 2.5 = 1.802775638

Simplifying
z + 2.5 = 1.802775638

Reorder the terms:
2.5 + z = 1.802775638

Solving
2.5 + z = 1.802775638

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 = 1.802775638 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + z = 1.802775638 + -2.5
z = 1.802775638 + -2.5

Combine like terms: 1.802775638 + -2.5 = -0.697224362
z = -0.697224362

Simplifying
z = -0.697224362

Subproblem 2
z + 2.5 = -1.802775638

Simplifying
z + 2.5 = -1.802775638

Reorder the terms:
2.5 + z = -1.802775638

Solving
2.5 + z = -1.802775638

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 = -1.802775638 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + z = -1.802775638 + -2.5
z = -1.802775638 + -2.5

Combine like terms: -1.802775638 + -2.5 = -4.302775638
z = -4.302775638

Simplifying
z = -4.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.697224362, -4.302775638}