3x^2+16x+6=0

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Solution for 3x^2+16x+6=0 equation: 


Simplifying
3x2 + 16x + 6 = 0

Reorder the terms:
6 + 16x + 3x2 = 0

Solving
6 + 16x + 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'.
2 + 5.333333333x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 5.333333333x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + 5.333333333x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 5.333333333x + x2 = 0 + -2
5.333333333x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
5.333333333x + x2 = -2

The x term is 5.333333333x.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333x + 7.111111113 + x2 = -2 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333x + x2 = -2 + 7.111111113

Combine like terms: -2 + 7.111111113 = 5.111111113
7.111111113 + 5.333333333x + x2 = 5.111111113

Factor a perfect square on the left side:
(x + 2.666666667)(x + 2.666666667) = 5.111111113

Calculate the square root of the right side: 2.260776661

Break this problem into two subproblems by setting 
(x + 2.666666667) equal to 2.260776661 and -2.260776661.


Subproblem 1
x + 2.666666667 = 2.260776661

Simplifying
x + 2.666666667 = 2.260776661

Reorder the terms:
2.666666667 + x = 2.260776661

Solving
2.666666667 + x = 2.260776661

Solving for variable 'x'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + x = 2.260776661 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + x = 2.260776661 + -2.666666667
x = 2.260776661 + -2.666666667

Combine like terms: 2.260776661 + -2.666666667 = -0.405890006
x = -0.405890006

Simplifying
x = -0.405890006


Subproblem 2
x + 2.666666667 = -2.260776661

Simplifying
x + 2.666666667 = -2.260776661

Reorder the terms:
2.666666667 + x = -2.260776661

Solving
2.666666667 + x = -2.260776661

Solving for variable 'x'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + x = -2.260776661 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + x = -2.260776661 + -2.666666667
x = -2.260776661 + -2.666666667

Combine like terms: -2.260776661 + -2.666666667 = -4.927443328
x = -4.927443328

Simplifying
x = -4.927443328


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.405890006, -4.927443328}

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