4x^2-54x+108=0

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Solution for 4x^2-54x+108=0 equation: 


Simplifying
4x2 + -54x + 108 = 0

Reorder the terms:
108 + -54x + 4x2 = 0

Solving
108 + -54x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(54 + -27x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(54 + -27x + 2x2)' equal to zero and attempt to solve:

Simplifying
54 + -27x + 2x2 = 0

Solving
54 + -27x + 2x2 = 0

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

Divide each side by '2'.
27 + -13.5x + x2 = 0

Move the constant term to the right:

Add '-27' to each side of the equation.
27 + -13.5x + -27 + x2 = 0 + -27

Reorder the terms:
27 + -27 + -13.5x + x2 = 0 + -27

Combine like terms: 27 + -27 = 0
0 + -13.5x + x2 = 0 + -27
-13.5x + x2 = 0 + -27

Combine like terms: 0 + -27 = -27
-13.5x + x2 = -27

The x term is -13.5x.  Take half its coefficient (-6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
-13.5x + 45.5625 + x2 = -27 + 45.5625

Reorder the terms:
45.5625 + -13.5x + x2 = -27 + 45.5625

Combine like terms: -27 + 45.5625 = 18.5625
45.5625 + -13.5x + x2 = 18.5625

Factor a perfect square on the left side:
(x + -6.75)(x + -6.75) = 18.5625

Calculate the square root of the right side: 4.308421985

Break this problem into two subproblems by setting 
(x + -6.75) equal to 4.308421985 and -4.308421985.


Subproblem 1
x + -6.75 = 4.308421985

Simplifying
x + -6.75 = 4.308421985

Reorder the terms:
-6.75 + x = 4.308421985

Solving
-6.75 + x = 4.308421985

Solving for variable 'x'.

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

Add '6.75' to each side of the equation.
-6.75 + 6.75 + x = 4.308421985 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x = 4.308421985 + 6.75
x = 4.308421985 + 6.75

Combine like terms: 4.308421985 + 6.75 = 11.058421985
x = 11.058421985

Simplifying
x = 11.058421985


Subproblem 2
x + -6.75 = -4.308421985

Simplifying
x + -6.75 = -4.308421985

Reorder the terms:
-6.75 + x = -4.308421985

Solving
-6.75 + x = -4.308421985

Solving for variable 'x'.

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

Add '6.75' to each side of the equation.
-6.75 + 6.75 + x = -4.308421985 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x = -4.308421985 + 6.75
x = -4.308421985 + 6.75

Combine like terms: -4.308421985 + 6.75 = 2.441578015
x = 2.441578015

Simplifying
x = 2.441578015


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.058421985, 2.441578015}
Solution
x = {11.058421985, 2.441578015}

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