3x^2+19x+27=0

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Solution for 3x^2+19x+27=0 equation: 


Simplifying
3x2 + 19x + 27 = 0

Reorder the terms:
27 + 19x + 3x2 = 0

Solving
27 + 19x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
9 + 6.333333333x + x2 = 0

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + 6.333333333x + -9 + x2 = 0 + -9

Reorder the terms:
9 + -9 + 6.333333333x + x2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + 6.333333333x + x2 = 0 + -9
6.333333333x + x2 = 0 + -9

Combine like terms: 0 + -9 = -9
6.333333333x + x2 = -9

The x term is 6.333333333x.  Take half its coefficient (3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
6.333333333x + 10.02777778 + x2 = -9 + 10.02777778

Reorder the terms:
10.02777778 + 6.333333333x + x2 = -9 + 10.02777778

Combine like terms: -9 + 10.02777778 = 1.02777778
10.02777778 + 6.333333333x + x2 = 1.02777778

Factor a perfect square on the left side:
(x + 3.166666667)(x + 3.166666667) = 1.02777778

Calculate the square root of the right side: 1.013793756

Break this problem into two subproblems by setting 
(x + 3.166666667) equal to 1.013793756 and -1.013793756.


Subproblem 1
x + 3.166666667 = 1.013793756

Simplifying
x + 3.166666667 = 1.013793756

Reorder the terms:
3.166666667 + x = 1.013793756

Solving
3.166666667 + x = 1.013793756

Solving for variable 'x'.

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

Add '-3.166666667' to each side of the equation.
3.166666667 + -3.166666667 + x = 1.013793756 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + x = 1.013793756 + -3.166666667
x = 1.013793756 + -3.166666667

Combine like terms: 1.013793756 + -3.166666667 = -2.152872911
x = -2.152872911

Simplifying
x = -2.152872911


Subproblem 2
x + 3.166666667 = -1.013793756

Simplifying
x + 3.166666667 = -1.013793756

Reorder the terms:
3.166666667 + x = -1.013793756

Solving
3.166666667 + x = -1.013793756

Solving for variable 'x'.

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

Add '-3.166666667' to each side of the equation.
3.166666667 + -3.166666667 + x = -1.013793756 + -3.166666667

Combine like terms: 3.166666667 + -3.166666667 = 0.000000000
0.000000000 + x = -1.013793756 + -3.166666667
x = -1.013793756 + -3.166666667

Combine like terms: -1.013793756 + -3.166666667 = -4.180460423
x = -4.180460423

Simplifying
x = -4.180460423


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.152872911, -4.180460423}

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