0.5x^2+2x+8=0.25x^2-x

Solution for 0.5x^2+2x+8=0.25x^2-x equation:

Simplifying
0.5x2 + 2x + 8 = 0.25x2 + -1x

Reorder the terms:
8 + 2x + 0.5x2 = 0.25x2 + -1x

Reorder the terms:
8 + 2x + 0.5x2 = -1x + 0.25x2

Solving
8 + 2x + 0.5x2 = -1x + 0.25x2

Solving for variable 'x'.

Reorder the terms:
8 + 2x + x + 0.5x2 + -0.25x2 = -1x + 0.25x2 + x + -0.25x2

Combine like terms: 2x + x = 3x
8 + 3x + 0.5x2 + -0.25x2 = -1x + 0.25x2 + x + -0.25x2

Combine like terms: 0.5x2 + -0.25x2 = 0.25x2
8 + 3x + 0.25x2 = -1x + 0.25x2 + x + -0.25x2

Reorder the terms:
8 + 3x + 0.25x2 = -1x + x + 0.25x2 + -0.25x2

Combine like terms: -1x + x = 0
8 + 3x + 0.25x2 = 0 + 0.25x2 + -0.25x2
8 + 3x + 0.25x2 = 0.25x2 + -0.25x2

Combine like terms: 0.25x2 + -0.25x2 = 0.00
8 + 3x + 0.25x2 = 0.00

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

Divide each side by '0.25'.
32 + 12x + x2 = 0

Move the constant term to the right:

Add '-32' to each side of the equation.
32 + 12x + -32 + x2 = 0 + -32

Reorder the terms:
32 + -32 + 12x + x2 = 0 + -32

Combine like terms: 32 + -32 = 0
0 + 12x + x2 = 0 + -32
12x + x2 = 0 + -32

Combine like terms: 0 + -32 = -32
12x + x2 = -32

The x term is 12x.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12x + 36 + x2 = -32 + 36

Reorder the terms:
36 + 12x + x2 = -32 + 36

Combine like terms: -32 + 36 = 4
36 + 12x + x2 = 4

Factor a perfect square on the left side:
(x + 6)(x + 6) = 4

Calculate the square root of the right side: 2

Break this problem into two subproblems by setting 
(x + 6) equal to 2 and -2.

Subproblem 1
x + 6 = 2

Simplifying
x + 6 = 2

Reorder the terms:
6 + x = 2

Solving
6 + x = 2

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = 2 + -6

Combine like terms: 6 + -6 = 0
0 + x = 2 + -6
x = 2 + -6

Combine like terms: 2 + -6 = -4
x = -4

Simplifying
x = -4

Subproblem 2
x + 6 = -2

Simplifying
x + 6 = -2

Reorder the terms:
6 + x = -2

Solving
6 + x = -2

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = -2 + -6

Combine like terms: 6 + -6 = 0
0 + x = -2 + -6
x = -2 + -6

Combine like terms: -2 + -6 = -8
x = -8

Simplifying
x = -8

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-4, -8}