9x^2-27x-6=0

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Solution for 9x^2-27x-6=0 equation: 


Simplifying
9x2 + -27x + -6 = 0

Reorder the terms:
-6 + -27x + 9x2 = 0

Solving
-6 + -27x + 9x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-2 + -9x + 3x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-2 + -9x + 3x2)' equal to zero and attempt to solve:

Simplifying
-2 + -9x + 3x2 = 0

Solving
-2 + -9x + 3x2 = 0

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

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

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -3x + 0.6666666667 + x2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -3x + x2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -3x + x2 = 0 + 0.6666666667
-3x + x2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-3x + x2 = 0.6666666667

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 0.6666666667 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 0.6666666667 + 2.25

Combine like terms: 0.6666666667 + 2.25 = 2.9166666667
2.25 + -3x + x2 = 2.9166666667

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 2.9166666667

Calculate the square root of the right side: 1.707825128

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.707825128 and -1.707825128.


Subproblem 1
x + -1.5 = 1.707825128

Simplifying
x + -1.5 = 1.707825128

Reorder the terms:
-1.5 + x = 1.707825128

Solving
-1.5 + x = 1.707825128

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 1.707825128 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 1.707825128 + 1.5
x = 1.707825128 + 1.5

Combine like terms: 1.707825128 + 1.5 = 3.207825128
x = 3.207825128

Simplifying
x = 3.207825128


Subproblem 2
x + -1.5 = -1.707825128

Simplifying
x + -1.5 = -1.707825128

Reorder the terms:
-1.5 + x = -1.707825128

Solving
-1.5 + x = -1.707825128

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -1.707825128 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -1.707825128 + 1.5
x = -1.707825128 + 1.5

Combine like terms: -1.707825128 + 1.5 = -0.207825128
x = -0.207825128

Simplifying
x = -0.207825128


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.207825128, -0.207825128}
Solution
x = {3.207825128, -0.207825128}

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