z^2+5z+1=0

Solution for z^2+5z+1=0 equation:

Simplifying
z2 + 5z + 1 = 0

Reorder the terms:
1 + 5z + z2 = 0

Solving
1 + 5z + 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 + 5z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 5z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 5z + z2 = 0 + -1
5z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
5z + z2 = -1

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 = -1 + 6.25

Reorder the terms:
6.25 + 5z + z2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + 5z + z2 = 5.25

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(z + 2.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
z + 2.5 = 2.291287847

Simplifying
z + 2.5 = 2.291287847

Reorder the terms:
2.5 + z = 2.291287847

Solving
2.5 + z = 2.291287847

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 = 2.291287847 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + z = 2.291287847 + -2.5
z = 2.291287847 + -2.5

Combine like terms: 2.291287847 + -2.5 = -0.208712153
z = -0.208712153

Simplifying
z = -0.208712153

Subproblem 2
z + 2.5 = -2.291287847

Simplifying
z + 2.5 = -2.291287847

Reorder the terms:
2.5 + z = -2.291287847

Solving
2.5 + z = -2.291287847

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 = -2.291287847 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + z = -2.291287847 + -2.5
z = -2.291287847 + -2.5

Combine like terms: -2.291287847 + -2.5 = -4.791287847
z = -4.791287847

Simplifying
z = -4.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.208712153, -4.791287847}