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 = 1.5833333333
2.25 + 3x + x2 = 1.5833333333

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 1.5833333333

Calculate the square root of the right side: 1.258305739

Break this problem into two subproblems by setting 
(x + 1.5) equal to 1.258305739 and -1.258305739.


Subproblem 1
x + 1.5 = 1.258305739

Simplifying
x + 1.5 = 1.258305739

Reorder the terms:
1.5 + x = 1.258305739

Solving
1.5 + x = 1.258305739

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 = 1.258305739 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 1.258305739 + -1.5
x = 1.258305739 + -1.5

Combine like terms: 1.258305739 + -1.5 = -0.241694261
x = -0.241694261

Simplifying
x = -0.241694261


Subproblem 2
x + 1.5 = -1.258305739

Simplifying
x + 1.5 = -1.258305739

Reorder the terms:
1.5 + x = -1.258305739

Solving
1.5 + x = -1.258305739

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 = -1.258305739 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -1.258305739 + -1.5
x = -1.258305739 + -1.5

Combine like terms: -1.258305739 + -1.5 = -2.758305739
x = -2.758305739

Simplifying
x = -2.758305739


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.241694261, -2.758305739}
Solution
x = {-0.241694261, -2.758305739}

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