.5x^2-x=12

Solution for .5x^2-x=12 equation:

Simplifying
0.5x2 + -1x = 12

Reorder the terms:
-1x + 0.5x2 = 12

Solving
-1x + 0.5x2 = 12

Solving for variable 'x'.

Reorder the terms:
-12 + -1x + 0.5x2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + -1x + 0.5x2 = 0

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

Divide each side by '0.5'.
-24 + -2x + x2 = 0

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + -2x + 24 + x2 = 0 + 24

Reorder the terms:
-24 + 24 + -2x + x2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + -2x + x2 = 0 + 24
-2x + x2 = 0 + 24

Combine like terms: 0 + 24 = 24
-2x + x2 = 24

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 24 + 1

Reorder the terms:
1 + -2x + x2 = 24 + 1

Combine like terms: 24 + 1 = 25
1 + -2x + x2 = 25

Factor a perfect square on the left side:
(x + -1)(x + -1) = 25

Calculate the square root of the right side: 5

Break this problem into two subproblems by setting 
(x + -1) equal to 5 and -5.

Subproblem 1
x + -1 = 5

Simplifying
x + -1 = 5

Reorder the terms:
-1 + x = 5

Solving
-1 + x = 5

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = 5 + 1

Combine like terms: -1 + 1 = 0
0 + x = 5 + 1
x = 5 + 1

Combine like terms: 5 + 1 = 6
x = 6

Simplifying
x = 6

Subproblem 2
x + -1 = -5

Simplifying
x + -1 = -5

Reorder the terms:
-1 + x = -5

Solving
-1 + x = -5

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = -5 + 1

Combine like terms: -1 + 1 = 0
0 + x = -5 + 1
x = -5 + 1

Combine like terms: -5 + 1 = -4
x = -4

Simplifying
x = -4

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6, -4}