6x(2x+1)-54=0

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Solution for 6x(2x+1)-54=0 equation: 


Simplifying
6x(2x + 1) + -54 = 0

Reorder the terms:
6x(1 + 2x) + -54 = 0
(1 * 6x + 2x * 6x) + -54 = 0
(6x + 12x2) + -54 = 0

Reorder the terms:
-54 + 6x + 12x2 = 0

Solving
-54 + 6x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(-9 + x + 2x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(-9 + x + 2x2)' equal to zero and attempt to solve:

Simplifying
-9 + x + 2x2 = 0

Solving
-9 + x + 2x2 = 0

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

Divide each side by '2'.
-4.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 0.5x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 0.5x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 0.5x + x2 = 0 + 4.5
0.5x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
0.5x + x2 = 4.5

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5x + 0.25 + x2 = 4.5 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 4.5 + 0.25

Combine like terms: 4.5 + 0.25 = 4.75
0.25 + 0.5x + x2 = 4.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 4.75

Calculate the square root of the right side: 2.179449472

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.179449472 and -2.179449472.


Subproblem 1
x + 0.5 = 2.179449472

Simplifying
x + 0.5 = 2.179449472

Reorder the terms:
0.5 + x = 2.179449472

Solving
0.5 + x = 2.179449472

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 2.179449472 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.179449472 + -0.5
x = 2.179449472 + -0.5

Combine like terms: 2.179449472 + -0.5 = 1.679449472
x = 1.679449472

Simplifying
x = 1.679449472


Subproblem 2
x + 0.5 = -2.179449472

Simplifying
x + 0.5 = -2.179449472

Reorder the terms:
0.5 + x = -2.179449472

Solving
0.5 + x = -2.179449472

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -2.179449472 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.179449472 + -0.5
x = -2.179449472 + -0.5

Combine like terms: -2.179449472 + -0.5 = -2.679449472
x = -2.679449472

Simplifying
x = -2.679449472


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.679449472, -2.679449472}
Solution
x = {1.679449472, -2.679449472}

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