3+7z+z^2=0

Solution for 3+7z+z^2=0 equation:

Simplifying
3 + 7z + z2 = 0

Solving
3 + 7z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 7z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + 7z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 7z + z2 = 0 + -3
7z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
7z + z2 = -3

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 = -3 + 12.25

Reorder the terms:
12.25 + 7z + z2 = -3 + 12.25

Combine like terms: -3 + 12.25 = 9.25
12.25 + 7z + z2 = 9.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(z + 3.5) equal to 3.041381265 and -3.041381265.

Subproblem 1
z + 3.5 = 3.041381265

Simplifying
z + 3.5 = 3.041381265

Reorder the terms:
3.5 + z = 3.041381265

Solving
3.5 + z = 3.041381265

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 = 3.041381265 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + z = 3.041381265 + -3.5
z = 3.041381265 + -3.5

Combine like terms: 3.041381265 + -3.5 = -0.458618735
z = -0.458618735

Simplifying
z = -0.458618735

Subproblem 2
z + 3.5 = -3.041381265

Simplifying
z + 3.5 = -3.041381265

Reorder the terms:
3.5 + z = -3.041381265

Solving
3.5 + z = -3.041381265

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 = -3.041381265 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + z = -3.041381265 + -3.5
z = -3.041381265 + -3.5

Combine like terms: -3.041381265 + -3.5 = -6.541381265
z = -6.541381265

Simplifying
z = -6.541381265

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.458618735, -6.541381265}