12x^2-54x+16=0

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Solution for 12x^2-54x+16=0 equation: 


Simplifying
12x2 + -54x + 16 = 0

Reorder the terms:
16 + -54x + 12x2 = 0

Solving
16 + -54x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(8 + -27x + 6x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(8 + -27x + 6x2)' equal to zero and attempt to solve:

Simplifying
8 + -27x + 6x2 = 0

Solving
8 + -27x + 6x2 = 0

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

Divide each side by '6'.
1.333333333 + -4.5x + x2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + -4.5x + -1.333333333 + x2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + -4.5x + x2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + -4.5x + x2 = 0 + -1.333333333
-4.5x + x2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
-4.5x + x2 = -1.333333333

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = -1.333333333 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = -1.333333333 + 5.0625

Combine like terms: -1.333333333 + 5.0625 = 3.729166667
5.0625 + -4.5x + x2 = 3.729166667

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 3.729166667

Calculate the square root of the right side: 1.931105038

Break this problem into two subproblems by setting 
(x + -2.25) equal to 1.931105038 and -1.931105038.


Subproblem 1
x + -2.25 = 1.931105038

Simplifying
x + -2.25 = 1.931105038

Reorder the terms:
-2.25 + x = 1.931105038

Solving
-2.25 + x = 1.931105038

Solving for variable 'x'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + x = 1.931105038 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = 1.931105038 + 2.25
x = 1.931105038 + 2.25

Combine like terms: 1.931105038 + 2.25 = 4.181105038
x = 4.181105038

Simplifying
x = 4.181105038


Subproblem 2
x + -2.25 = -1.931105038

Simplifying
x + -2.25 = -1.931105038

Reorder the terms:
-2.25 + x = -1.931105038

Solving
-2.25 + x = -1.931105038

Solving for variable 'x'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + x = -1.931105038 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + x = -1.931105038 + 2.25
x = -1.931105038 + 2.25

Combine like terms: -1.931105038 + 2.25 = 0.318894962
x = 0.318894962

Simplifying
x = 0.318894962


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.181105038, 0.318894962}
Solution
x = {4.181105038, 0.318894962}

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