z^2+7z=1

Solution for z^2+7z=1 equation:

Simplifying
z2 + 7z = 1

Reorder the terms:
7z + z2 = 1

Solving
7z + z2 = 1

Solving for variable 'z'.

Reorder the terms:
-1 + 7z + z2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 7z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 7z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 7z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 7z + z2 = 0 + 1
7z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
7z + z2 = 1

The z term is 7z.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7z + 12.25 + z2 = 1 + 12.25

Reorder the terms:
12.25 + 7z + z2 = 1 + 12.25

Combine like terms: 1 + 12.25 = 13.25
12.25 + 7z + z2 = 13.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(z + 3.5) equal to 3.640054945 and -3.640054945.

Subproblem 1
z + 3.5 = 3.640054945

Simplifying
z + 3.5 = 3.640054945

Reorder the terms:
3.5 + z = 3.640054945

Solving
3.5 + z = 3.640054945

Solving for variable 'z'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + z = 3.640054945 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + z = 3.640054945 + -3.5
z = 3.640054945 + -3.5

Combine like terms: 3.640054945 + -3.5 = 0.140054945
z = 0.140054945

Simplifying
z = 0.140054945

Subproblem 2
z + 3.5 = -3.640054945

Simplifying
z + 3.5 = -3.640054945

Reorder the terms:
3.5 + z = -3.640054945

Solving
3.5 + z = -3.640054945

Solving for variable 'z'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + z = -3.640054945 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + z = -3.640054945 + -3.5
z = -3.640054945 + -3.5

Combine like terms: -3.640054945 + -3.5 = -7.140054945
z = -7.140054945

Simplifying
z = -7.140054945

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.140054945, -7.140054945}