0.25z^2+z-0.5=0

Solution for 0.25z^2+z-0.5=0 equation:

Simplifying
0.25z2 + z + -0.5 = 0

Reorder the terms:
-0.5 + z + 0.25z2 = 0

Solving
-0.5 + z + 0.25z2 = 0

Solving for variable 'z'.

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

Divide each side by '0.25'.
-2 + 4z + z2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 4z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + 4z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 4z + z2 = 0 + 2
4z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
4z + z2 = 2

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.
4z + 0.25 + z2 = 2 + 0.25

Reorder the terms:
0.25 + 4z + z2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
0.25 + 4z + z2 = 2.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(z + 0.5) equal to 1.5 and -1.5.

Subproblem 1
z + 0.5 = 1.5

Simplifying
z + 0.5 = 1.5

Reorder the terms:
0.5 + z = 1.5

Solving
0.5 + z = 1.5

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 = 1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 1.5 + -0.5
z = 1.5 + -0.5

Combine like terms: 1.5 + -0.5 = 1
z = 1

Simplifying
z = 1

Subproblem 2
z + 0.5 = -1.5

Simplifying
z + 0.5 = -1.5

Reorder the terms:
0.5 + z = -1.5

Solving
0.5 + z = -1.5

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 = -1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -1.5 + -0.5
z = -1.5 + -0.5

Combine like terms: -1.5 + -0.5 = -2
z = -2

Simplifying
z = -2

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1, -2}