z^2+5z=1

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

Simplifying
z2 + 5z = 1

Reorder the terms:
5z + z2 = 1

Solving
5z + z2 = 1

Solving for variable 'z'.

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

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

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 = 7.25
6.25 + 5z + z2 = 7.25

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(z + 2.5) equal to 2.692582404 and -2.692582404.

Subproblem 1
z + 2.5 = 2.692582404

Simplifying
z + 2.5 = 2.692582404

Reorder the terms:
2.5 + z = 2.692582404

Solving
2.5 + z = 2.692582404

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 = 2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + z = 2.692582404 + -2.5
z = 2.692582404 + -2.5

Combine like terms: 2.692582404 + -2.5 = 0.192582404
z = 0.192582404

Simplifying
z = 0.192582404

Subproblem 2
z + 2.5 = -2.692582404

Simplifying
z + 2.5 = -2.692582404

Reorder the terms:
2.5 + z = -2.692582404

Solving
2.5 + z = -2.692582404

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 = -2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + z = -2.692582404 + -2.5
z = -2.692582404 + -2.5

Combine like terms: -2.692582404 + -2.5 = -5.192582404
z = -5.192582404

Simplifying
z = -5.192582404

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.192582404, -5.192582404}