3x-12=2x+8-6x^2-16

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Solution for 3x-12=2x+8-6x^2-16 equation: 


Simplifying
3x + -12 = 2x + 8 + -6x2 + -16

Reorder the terms:
-12 + 3x = 2x + 8 + -6x2 + -16

Reorder the terms:
-12 + 3x = 8 + -16 + 2x + -6x2

Combine like terms: 8 + -16 = -8
-12 + 3x = -8 + 2x + -6x2

Solving
-12 + 3x = -8 + 2x + -6x2

Solving for variable 'x'.

Reorder the terms:
-12 + 8 + 3x + -2x + 6x2 = -8 + 2x + -6x2 + 8 + -2x + 6x2

Combine like terms: -12 + 8 = -4
-4 + 3x + -2x + 6x2 = -8 + 2x + -6x2 + 8 + -2x + 6x2

Combine like terms: 3x + -2x = 1x
-4 + 1x + 6x2 = -8 + 2x + -6x2 + 8 + -2x + 6x2

Reorder the terms:
-4 + 1x + 6x2 = -8 + 8 + 2x + -2x + -6x2 + 6x2

Combine like terms: -8 + 8 = 0
-4 + 1x + 6x2 = 0 + 2x + -2x + -6x2 + 6x2
-4 + 1x + 6x2 = 2x + -2x + -6x2 + 6x2

Combine like terms: 2x + -2x = 0
-4 + 1x + 6x2 = 0 + -6x2 + 6x2
-4 + 1x + 6x2 = -6x2 + 6x2

Combine like terms: -6x2 + 6x2 = 0
-4 + 1x + 6x2 = 0

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

Divide each side by '6'.
-0.6666666667 + 0.1666666667x + x2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + 0.1666666667x + 0.6666666667 + x2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + 0.1666666667x + x2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + 0.1666666667x + x2 = 0 + 0.6666666667
0.1666666667x + x2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
0.1666666667x + x2 = 0.6666666667

The x term is 0.1666666667x.  Take half its coefficient (0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
0.1666666667x + 0.006944444447 + x2 = 0.6666666667 + 0.006944444447

Reorder the terms:
0.006944444447 + 0.1666666667x + x2 = 0.6666666667 + 0.006944444447

Combine like terms: 0.6666666667 + 0.006944444447 = 0.673611111147
0.006944444447 + 0.1666666667x + x2 = 0.673611111147

Factor a perfect square on the left side:
(x + 0.08333333335)(x + 0.08333333335) = 0.673611111147

Calculate the square root of the right side: 0.82073815

Break this problem into two subproblems by setting 
(x + 0.08333333335) equal to 0.82073815 and -0.82073815.


Subproblem 1
x + 0.08333333335 = 0.82073815

Simplifying
x + 0.08333333335 = 0.82073815

Reorder the terms:
0.08333333335 + x = 0.82073815

Solving
0.08333333335 + x = 0.82073815

Solving for variable 'x'.

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

Add '-0.08333333335' to each side of the equation.
0.08333333335 + -0.08333333335 + x = 0.82073815 + -0.08333333335

Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000
0.00000000000 + x = 0.82073815 + -0.08333333335
x = 0.82073815 + -0.08333333335

Combine like terms: 0.82073815 + -0.08333333335 = 0.73740481665
x = 0.73740481665

Simplifying
x = 0.73740481665


Subproblem 2
x + 0.08333333335 = -0.82073815

Simplifying
x + 0.08333333335 = -0.82073815

Reorder the terms:
0.08333333335 + x = -0.82073815

Solving
0.08333333335 + x = -0.82073815

Solving for variable 'x'.

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

Add '-0.08333333335' to each side of the equation.
0.08333333335 + -0.08333333335 + x = -0.82073815 + -0.08333333335

Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000
0.00000000000 + x = -0.82073815 + -0.08333333335
x = -0.82073815 + -0.08333333335

Combine like terms: -0.82073815 + -0.08333333335 = -0.90407148335
x = -0.90407148335

Simplifying
x = -0.90407148335


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.73740481665, -0.90407148335}

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