z^2+10z+6=0

Solution for z^2+10z+6=0 equation:

Simplifying
z2 + 10z + 6 = 0

Reorder the terms:
6 + 10z + z2 = 0

Solving
6 + 10z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + 10z + -6 + z2 = 0 + -6

Reorder the terms:
6 + -6 + 10z + z2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + 10z + z2 = 0 + -6
10z + z2 = 0 + -6

Combine like terms: 0 + -6 = -6
10z + z2 = -6

The z term is 10z.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10z + 25 + z2 = -6 + 25

Reorder the terms:
25 + 10z + z2 = -6 + 25

Combine like terms: -6 + 25 = 19
25 + 10z + z2 = 19

Factor a perfect square on the left side:
(z + 5)(z + 5) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(z + 5) equal to 4.358898944 and -4.358898944.

Subproblem 1
z + 5 = 4.358898944

Simplifying
z + 5 = 4.358898944

Reorder the terms:
5 + z = 4.358898944

Solving
5 + z = 4.358898944

Solving for variable 'z'.

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

Add '-5' to each side of the equation.
5 + -5 + z = 4.358898944 + -5

Combine like terms: 5 + -5 = 0
0 + z = 4.358898944 + -5
z = 4.358898944 + -5

Combine like terms: 4.358898944 + -5 = -0.641101056
z = -0.641101056

Simplifying
z = -0.641101056

Subproblem 2
z + 5 = -4.358898944

Simplifying
z + 5 = -4.358898944

Reorder the terms:
5 + z = -4.358898944

Solving
5 + z = -4.358898944

Solving for variable 'z'.

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

Add '-5' to each side of the equation.
5 + -5 + z = -4.358898944 + -5

Combine like terms: 5 + -5 = 0
0 + z = -4.358898944 + -5
z = -4.358898944 + -5

Combine like terms: -4.358898944 + -5 = -9.358898944
z = -9.358898944

Simplifying
z = -9.358898944

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.641101056, -9.358898944}