12x-x^2=36-12x+x^2

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Solution for 12x-x^2=36-12x+x^2 equation: 


Simplifying
12x + -1x2 = 36 + -12x + x2

Solving
12x + -1x2 = 36 + -12x + x2

Solving for variable 'x'.

Reorder the terms:
-36 + 12x + 12x + -1x2 + -1x2 = 36 + -12x + x2 + -36 + 12x + -1x2

Combine like terms: 12x + 12x = 24x
-36 + 24x + -1x2 + -1x2 = 36 + -12x + x2 + -36 + 12x + -1x2

Combine like terms: -1x2 + -1x2 = -2x2
-36 + 24x + -2x2 = 36 + -12x + x2 + -36 + 12x + -1x2

Reorder the terms:
-36 + 24x + -2x2 = 36 + -36 + -12x + 12x + x2 + -1x2

Combine like terms: 36 + -36 = 0
-36 + 24x + -2x2 = 0 + -12x + 12x + x2 + -1x2
-36 + 24x + -2x2 = -12x + 12x + x2 + -1x2

Combine like terms: -12x + 12x = 0
-36 + 24x + -2x2 = 0 + x2 + -1x2
-36 + 24x + -2x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-36 + 24x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-18 + 12x + -1x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-18 + 12x + -1x2)' equal to zero and attempt to solve:

Simplifying
-18 + 12x + -1x2 = 0

Solving
-18 + 12x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
18 + -12x + x2 = 0

Move the constant term to the right:

Add '-18' to each side of the equation.
18 + -12x + -18 + x2 = 0 + -18

Reorder the terms:
18 + -18 + -12x + x2 = 0 + -18

Combine like terms: 18 + -18 = 0
0 + -12x + x2 = 0 + -18
-12x + x2 = 0 + -18

Combine like terms: 0 + -18 = -18
-12x + x2 = -18

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 = -18 + 36

Reorder the terms:
36 + -12x + x2 = -18 + 36

Combine like terms: -18 + 36 = 18
36 + -12x + x2 = 18

Factor a perfect square on the left side:
(x + -6)(x + -6) = 18

Calculate the square root of the right side: 4.242640687

Break this problem into two subproblems by setting 
(x + -6) equal to 4.242640687 and -4.242640687.


Subproblem 1
x + -6 = 4.242640687

Simplifying
x + -6 = 4.242640687

Reorder the terms:
-6 + x = 4.242640687

Solving
-6 + x = 4.242640687

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 = 4.242640687 + 6

Combine like terms: -6 + 6 = 0
0 + x = 4.242640687 + 6
x = 4.242640687 + 6

Combine like terms: 4.242640687 + 6 = 10.242640687
x = 10.242640687

Simplifying
x = 10.242640687


Subproblem 2
x + -6 = -4.242640687

Simplifying
x + -6 = -4.242640687

Reorder the terms:
-6 + x = -4.242640687

Solving
-6 + x = -4.242640687

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 = -4.242640687 + 6

Combine like terms: -6 + 6 = 0
0 + x = -4.242640687 + 6
x = -4.242640687 + 6

Combine like terms: -4.242640687 + 6 = 1.757359313
x = 1.757359313

Simplifying
x = 1.757359313


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {10.242640687, 1.757359313}
Solution
x = {10.242640687, 1.757359313}

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