6x^2-4x-54=0

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Solution for 6x^2-4x-54=0 equation: 


Simplifying
6x2 + -4x + -54 = 0

Reorder the terms:
-54 + -4x + 6x2 = 0

Solving
-54 + -4x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-27 + -2x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-27 + -2x + 3x2)' equal to zero and attempt to solve:

Simplifying
-27 + -2x + 3x2 = 0

Solving
-27 + -2x + 3x2 = 0

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

Divide each side by '3'.
-9 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + -0.6666666667x + 9 + x2 = 0 + 9

Reorder the terms:
-9 + 9 + -0.6666666667x + x2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -0.6666666667x + x2 = 0 + 9
-0.6666666667x + x2 = 0 + 9

Combine like terms: 0 + 9 = 9
-0.6666666667x + x2 = 9

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 9 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 9 + 0.1111111112

Combine like terms: 9 + 0.1111111112 = 9.1111111112
0.1111111112 + -0.6666666667x + x2 = 9.1111111112

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 9.1111111112

Calculate the square root of the right side: 3.018461713

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 3.018461713 and -3.018461713.


Subproblem 1
x + -0.3333333334 = 3.018461713

Simplifying
x + -0.3333333334 = 3.018461713

Reorder the terms:
-0.3333333334 + x = 3.018461713

Solving
-0.3333333334 + x = 3.018461713

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = 3.018461713 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 3.018461713 + 0.3333333334
x = 3.018461713 + 0.3333333334

Combine like terms: 3.018461713 + 0.3333333334 = 3.3517950464
x = 3.3517950464

Simplifying
x = 3.3517950464


Subproblem 2
x + -0.3333333334 = -3.018461713

Simplifying
x + -0.3333333334 = -3.018461713

Reorder the terms:
-0.3333333334 + x = -3.018461713

Solving
-0.3333333334 + x = -3.018461713

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = -3.018461713 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -3.018461713 + 0.3333333334
x = -3.018461713 + 0.3333333334

Combine like terms: -3.018461713 + 0.3333333334 = -2.6851283796
x = -2.6851283796

Simplifying
x = -2.6851283796


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.3517950464, -2.6851283796}
Solution
x = {3.3517950464, -2.6851283796}

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