z^2+4z+1=0

Solution for z^2+4z+1=0 equation:

Simplifying
z2 + 4z + 1 = 0

Reorder the terms:
1 + 4z + z2 = 0

Solving
1 + 4z + 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 + 4z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 4z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 4z + z2 = 0 + -1
4z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
4z + z2 = -1

The z term is 4z.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4z + 4 + z2 = -1 + 4

Reorder the terms:
4 + 4z + z2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + 4z + z2 = 3

Factor a perfect square on the left side:
(z + 2)(z + 2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(z + 2) equal to 1.732050808 and -1.732050808.

Subproblem 1
z + 2 = 1.732050808

Simplifying
z + 2 = 1.732050808

Reorder the terms:
2 + z = 1.732050808

Solving
2 + z = 1.732050808

Solving for variable 'z'.

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

Add '-2' to each side of the equation.
2 + -2 + z = 1.732050808 + -2

Combine like terms: 2 + -2 = 0
0 + z = 1.732050808 + -2
z = 1.732050808 + -2

Combine like terms: 1.732050808 + -2 = -0.267949192
z = -0.267949192

Simplifying
z = -0.267949192

Subproblem 2
z + 2 = -1.732050808

Simplifying
z + 2 = -1.732050808

Reorder the terms:
2 + z = -1.732050808

Solving
2 + z = -1.732050808

Solving for variable 'z'.

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

Add '-2' to each side of the equation.
2 + -2 + z = -1.732050808 + -2

Combine like terms: 2 + -2 = 0
0 + z = -1.732050808 + -2
z = -1.732050808 + -2

Combine like terms: -1.732050808 + -2 = -3.732050808
z = -3.732050808

Simplifying
z = -3.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.267949192, -3.732050808}