z^2+9z+1=0

Solution for z^2+9z+1=0 equation:

Simplifying
z2 + 9z + 1 = 0

Reorder the terms:
1 + 9z + z2 = 0

Solving
1 + 9z + 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 + 9z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 9z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 9z + z2 = 0 + -1
9z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
9z + z2 = -1

The z term is 9z.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9z + 20.25 + z2 = -1 + 20.25

Reorder the terms:
20.25 + 9z + z2 = -1 + 20.25

Combine like terms: -1 + 20.25 = 19.25
20.25 + 9z + z2 = 19.25

Factor a perfect square on the left side:
(z + 4.5)(z + 4.5) = 19.25

Calculate the square root of the right side: 4.387482194

Break this problem into two subproblems by setting 
(z + 4.5) equal to 4.387482194 and -4.387482194.

Subproblem 1
z + 4.5 = 4.387482194

Simplifying
z + 4.5 = 4.387482194

Reorder the terms:
4.5 + z = 4.387482194

Solving
4.5 + z = 4.387482194

Solving for variable 'z'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + z = 4.387482194 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + z = 4.387482194 + -4.5
z = 4.387482194 + -4.5

Combine like terms: 4.387482194 + -4.5 = -0.112517806
z = -0.112517806

Simplifying
z = -0.112517806

Subproblem 2
z + 4.5 = -4.387482194

Simplifying
z + 4.5 = -4.387482194

Reorder the terms:
4.5 + z = -4.387482194

Solving
4.5 + z = -4.387482194

Solving for variable 'z'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + z = -4.387482194 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + z = -4.387482194 + -4.5
z = -4.387482194 + -4.5

Combine like terms: -4.387482194 + -4.5 = -8.887482194
z = -8.887482194

Simplifying
z = -8.887482194

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.112517806, -8.887482194}