0.5x^2-0.5x-1=0

Solution for 0.5x^2-0.5x-1=0 equation:

Simplifying
0.5x2 + -0.5x + -1 = 0

Reorder the terms:
-1 + -0.5x + 0.5x2 = 0

Solving
-1 + -0.5x + 0.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.5'.
-2 + -1x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -1x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1x + x2 = 0 + 2
-1x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1x + x2 = 2

The x term is -1x.  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.
-1x + 0.25 + x2 = 2 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
0.25 + -1x + x2 = 2.25

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

Calculate the square root of the right side: 1.5

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

Subproblem 1
x + -0.5 = 1.5

Simplifying
x + -0.5 = 1.5

Reorder the terms:
-0.5 + x = 1.5

Solving
-0.5 + x = 1.5

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = 1.5 + 0.5

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

Combine like terms: 1.5 + 0.5 = 2
x = 2

Simplifying
x = 2

Subproblem 2
x + -0.5 = -1.5

Simplifying
x + -0.5 = -1.5

Reorder the terms:
-0.5 + x = -1.5

Solving
-0.5 + x = -1.5

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = -1.5 + 0.5

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

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

Simplifying
x = -1

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