2x^2+27x+9=0

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Solution for 2x^2+27x+9=0 equation: 


Simplifying
2x2 + 27x + 9 = 0

Reorder the terms:
9 + 27x + 2x2 = 0

Solving
9 + 27x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
4.5 + 13.5x + x2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + 13.5x + -4.5 + x2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + 13.5x + x2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + 13.5x + x2 = 0 + -4.5
13.5x + x2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
13.5x + x2 = -4.5

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 = -4.5 + 45.5625

Reorder the terms:
45.5625 + 13.5x + x2 = -4.5 + 45.5625

Combine like terms: -4.5 + 45.5625 = 41.0625
45.5625 + 13.5x + x2 = 41.0625

Factor a perfect square on the left side:
(x + 6.75)(x + 6.75) = 41.0625

Calculate the square root of the right side: 6.408002809

Break this problem into two subproblems by setting 
(x + 6.75) equal to 6.408002809 and -6.408002809.


Subproblem 1
x + 6.75 = 6.408002809

Simplifying
x + 6.75 = 6.408002809

Reorder the terms:
6.75 + x = 6.408002809

Solving
6.75 + x = 6.408002809

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 = 6.408002809 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + x = 6.408002809 + -6.75
x = 6.408002809 + -6.75

Combine like terms: 6.408002809 + -6.75 = -0.341997191
x = -0.341997191

Simplifying
x = -0.341997191


Subproblem 2
x + 6.75 = -6.408002809

Simplifying
x + 6.75 = -6.408002809

Reorder the terms:
6.75 + x = -6.408002809

Solving
6.75 + x = -6.408002809

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 = -6.408002809 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + x = -6.408002809 + -6.75
x = -6.408002809 + -6.75

Combine like terms: -6.408002809 + -6.75 = -13.158002809
x = -13.158002809

Simplifying
x = -13.158002809


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.341997191, -13.158002809}

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