(z^2+z)+(z^2-11)=0

Solution for (z^2+z)+(z^2-11)=0 equation:

Simplifying
(z2 + z) + (z2 + -11) = 0

Reorder the terms:
(z + z2) + (z2 + -11) = 0

Remove parenthesis around (z + z2)
z + z2 + (z2 + -11) = 0

Reorder the terms:
z + z2 + (-11 + z2) = 0

Remove parenthesis around (-11 + z2)
z + z2 + -11 + z2 = 0

Reorder the terms:
-11 + z + z2 + z2 = 0

Combine like terms: z2 + z2 = 2z2
-11 + z + 2z2 = 0

Solving
-11 + z + 2z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-5.5 + 0.5z + z2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + 0.5z + 5.5 + z2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + 0.5z + z2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + 0.5z + z2 = 0 + 5.5
0.5z + z2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
0.5z + z2 = 5.5

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5z + 0.25 + z2 = 5.5 + 0.25

Reorder the terms:
0.25 + 0.5z + z2 = 5.5 + 0.25

Combine like terms: 5.5 + 0.25 = 5.75
0.25 + 0.5z + z2 = 5.75

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 5.75

Calculate the square root of the right side: 2.397915762

Break this problem into two subproblems by setting 
(z + 0.5) equal to 2.397915762 and -2.397915762.

Subproblem 1
z + 0.5 = 2.397915762

Simplifying
z + 0.5 = 2.397915762

Reorder the terms:
0.5 + z = 2.397915762

Solving
0.5 + z = 2.397915762

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z = 2.397915762 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 2.397915762 + -0.5
z = 2.397915762 + -0.5

Combine like terms: 2.397915762 + -0.5 = 1.897915762
z = 1.897915762

Simplifying
z = 1.897915762

Subproblem 2
z + 0.5 = -2.397915762

Simplifying
z + 0.5 = -2.397915762

Reorder the terms:
0.5 + z = -2.397915762

Solving
0.5 + z = -2.397915762

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z = -2.397915762 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -2.397915762 + -0.5
z = -2.397915762 + -0.5

Combine like terms: -2.397915762 + -0.5 = -2.897915762
z = -2.897915762

Simplifying
z = -2.897915762

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.897915762, -2.897915762}