z^2+z+0.25=0

Solution for z^2+z+0.25=0 equation:

Simplifying
z2 + z + 0.25 = 0

Reorder the terms:
0.25 + z + z2 = 0

Solving
0.25 + z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + z + -0.25 + z2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + z + z2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + z + z2 = 0 + -0.25
z + z2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
z + z2 = -0.25

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.
z + 0.25 + z2 = -0.25 + 0.25

Reorder the terms:
0.25 + z + z2 = -0.25 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.25 + z + z2 = 0.00

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 0.00

Calculate the square root of the right side: 0.00

Break this problem into two subproblems by setting 
(z + 0.5) equal to 0.00 and 0.00.

Subproblem 1
z + 0.5 = 0.00

Simplifying
z + 0.5 = 0.00

Reorder the terms:
0.5 + z = 0.00

Solving
0.5 + z = 0.00

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 = 0.00 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 0.00 + -0.5
z = 0.00 + -0.5

Combine like terms: 0.00 + -0.5 = -0.5
z = -0.5

Simplifying
z = -0.5

Subproblem 2
z + 0.5 = 0.00

Simplifying
z + 0.5 = 0.00

Reorder the terms:
0.5 + z = 0.00

Solving
0.5 + z = 0.00

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 = 0.00 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 0.00 + -0.5
z = 0.00 + -0.5

Combine like terms: 0.00 + -0.5 = -0.5
z = -0.5

Simplifying
z = -0.5

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.5, -0.5}