.5x^2+.5x-36=0

Solution for .5x^2+.5x-36=0 equation:

Simplifying
0.5x2 + 0.5x + -36 = 0

Reorder the terms:
-36 + 0.5x + 0.5x2 = 0

Solving
-36 + 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'.
-72 + x + x2 = 0

Move the constant term to the right:

Add '72' to each side of the equation.
-72 + x + 72 + x2 = 0 + 72

Reorder the terms:
-72 + 72 + x + x2 = 0 + 72

Combine like terms: -72 + 72 = 0
0 + x + x2 = 0 + 72
x + x2 = 0 + 72

Combine like terms: 0 + 72 = 72
x + x2 = 72

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 = 72 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 72 + 0.25

Combine like terms: 72 + 0.25 = 72.25
1.25 + x2 = 72.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 72.25

Calculate the square root of the right side: 8.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.5 and -8.5.

Subproblem 1
x + 0.5 = 8.5

Simplifying
x + 0.5 = 8.5

Reorder the terms:
0.5 + x = 8.5

Solving
0.5 + x = 8.5

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 = 8.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 8.5 + -0.5
x = 8.5 + -0.5

Combine like terms: 8.5 + -0.5 = 8
x = 8

Simplifying
x = 8

Subproblem 2
x + 0.5 = -8.5

Simplifying
x + 0.5 = -8.5

Reorder the terms:
0.5 + x = -8.5

Solving
0.5 + x = -8.5

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 = -8.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -8.5 + -0.5
x = -8.5 + -0.5

Combine like terms: -8.5 + -0.5 = -9
x = -9

Simplifying
x = -9

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8, -9}