2+7z+z^2=0

Solution for 2+7z+z^2=0 equation:

Simplifying
2 + 7z + z2 = 0

Solving
2 + 7z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 7z + -2 + z2 = 0 + -2

Reorder the terms:
2 + -2 + 7z + z2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 7z + z2 = 0 + -2
7z + z2 = 0 + -2

Combine like terms: 0 + -2 = -2
7z + z2 = -2

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 = -2 + 12.25

Reorder the terms:
12.25 + 7z + z2 = -2 + 12.25

Combine like terms: -2 + 12.25 = 10.25
12.25 + 7z + z2 = 10.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(z + 3.5) equal to 3.201562119 and -3.201562119.

Subproblem 1
z + 3.5 = 3.201562119

Simplifying
z + 3.5 = 3.201562119

Reorder the terms:
3.5 + z = 3.201562119

Solving
3.5 + z = 3.201562119

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 = 3.201562119 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + z = 3.201562119 + -3.5
z = 3.201562119 + -3.5

Combine like terms: 3.201562119 + -3.5 = -0.298437881
z = -0.298437881

Simplifying
z = -0.298437881

Subproblem 2
z + 3.5 = -3.201562119

Simplifying
z + 3.5 = -3.201562119

Reorder the terms:
3.5 + z = -3.201562119

Solving
3.5 + z = -3.201562119

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 = -3.201562119 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + z = -3.201562119 + -3.5
z = -3.201562119 + -3.5

Combine like terms: -3.201562119 + -3.5 = -6.701562119
z = -6.701562119

Simplifying
z = -6.701562119

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.298437881, -6.701562119}