19.77621629=z^2+z

Solution for 19.77621629=z^2+z equation:

Simplifying
19.77621629 = z2 + z

Reorder the terms:
19.77621629 = z + z2

Solving
19.77621629 = z + z2

Solving for variable 'z'.

Reorder the terms:
19.77621629 + -1z + -1z2 = z + -1z + z2 + -1z2

Combine like terms: z + -1z = 0
19.77621629 + -1z + -1z2 = 0 + z2 + -1z2
19.77621629 + -1z + -1z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
19.77621629 + -1z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-19.77621629 + z + z2 = 0

Move the constant term to the right:

Add '19.77621629' to each side of the equation.
-19.77621629 + z + 19.77621629 + z2 = 0 + 19.77621629

Reorder the terms:
-19.77621629 + 19.77621629 + z + z2 = 0 + 19.77621629

Combine like terms: -19.77621629 + 19.77621629 = 0.00000000
0.00000000 + z + z2 = 0 + 19.77621629
z + z2 = 0 + 19.77621629

Combine like terms: 0 + 19.77621629 = 19.77621629
z + z2 = 19.77621629

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.
 + 0.25 + z2 = 19.77621629 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + z2 = 19.77621629 + 0.25

Combine like terms: 19.77621629 + 0.25 = 20.02621629
1.25 + z2 = 20.02621629

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 20.02621629

Calculate the square root of the right side: 4.475066065

Break this problem into two subproblems by setting 
(z + 0.5) equal to 4.475066065 and -4.475066065.

Subproblem 1
z + 0.5 = 4.475066065

Simplifying
z + 0.5 = 4.475066065

Reorder the terms:
0.5 + z = 4.475066065

Solving
0.5 + z = 4.475066065

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 = 4.475066065 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 4.475066065 + -0.5
z = 4.475066065 + -0.5

Combine like terms: 4.475066065 + -0.5 = 3.975066065
z = 3.975066065

Simplifying
z = 3.975066065

Subproblem 2
z + 0.5 = -4.475066065

Simplifying
z + 0.5 = -4.475066065

Reorder the terms:
0.5 + z = -4.475066065

Solving
0.5 + z = -4.475066065

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 = -4.475066065 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -4.475066065 + -0.5
z = -4.475066065 + -0.5

Combine like terms: -4.475066065 + -0.5 = -4.975066065
z = -4.975066065

Simplifying
z = -4.975066065

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.975066065, -4.975066065}