-.5x^2-.5x+6=0

Solution for -.5x^2-.5x+6=0 equation:

Simplifying
-0.5x2 + -0.5x + 6 = 0

Reorder the terms:
6 + -0.5x + -0.5x2 = 0

Solving
6 + -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'.
-12 + x + x2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + x + 12 + x2 = 0 + 12

Reorder the terms:
-12 + 12 + x + x2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + x + x2 = 0 + 12
x + x2 = 0 + 12

Combine like terms: 0 + 12 = 12
x + x2 = 12

The x term is x.  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 + x2 = 12 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 12 + 0.25

Combine like terms: 12 + 0.25 = 12.25
1.25 + x2 = 12.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 12.25

Calculate the square root of the right side: 3.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.5 and -3.5.

Subproblem 1
x + 0.5 = 3.5

Simplifying
x + 0.5 = 3.5

Reorder the terms:
0.5 + x = 3.5

Solving
0.5 + x = 3.5

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 = 3.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 3.5 + -0.5
x = 3.5 + -0.5

Combine like terms: 3.5 + -0.5 = 3
x = 3

Simplifying
x = 3

Subproblem 2
x + 0.5 = -3.5

Simplifying
x + 0.5 = -3.5

Reorder the terms:
0.5 + x = -3.5

Solving
0.5 + x = -3.5

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 = -3.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -3.5 + -0.5
x = -3.5 + -0.5

Combine like terms: -3.5 + -0.5 = -4
x = -4

Simplifying
x = -4

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3, -4}