6x^2+24x+16=0

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Solution for 6x^2+24x+16=0 equation: 


Simplifying
6x2 + 24x + 16 = 0

Reorder the terms:
16 + 24x + 6x2 = 0

Solving
16 + 24x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(8 + 12x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(8 + 12x + 3x2)' equal to zero and attempt to solve:

Simplifying
8 + 12x + 3x2 = 0

Solving
8 + 12x + 3x2 = 0

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

Divide each side by '3'.
2.666666667 + 4x + x2 = 0

Move the constant term to the right:

Add '-2.666666667' to each side of the equation.
2.666666667 + 4x + -2.666666667 + x2 = 0 + -2.666666667

Reorder the terms:
2.666666667 + -2.666666667 + 4x + x2 = 0 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + 4x + x2 = 0 + -2.666666667
4x + x2 = 0 + -2.666666667

Combine like terms: 0 + -2.666666667 = -2.666666667
4x + x2 = -2.666666667

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = -2.666666667 + 4

Reorder the terms:
4 + 4x + x2 = -2.666666667 + 4

Combine like terms: -2.666666667 + 4 = 1.333333333
4 + 4x + x2 = 1.333333333

Factor a perfect square on the left side:
(x + 2)(x + 2) = 1.333333333

Calculate the square root of the right side: 1.154700538

Break this problem into two subproblems by setting 
(x + 2) equal to 1.154700538 and -1.154700538.


Subproblem 1
x + 2 = 1.154700538

Simplifying
x + 2 = 1.154700538

Reorder the terms:
2 + x = 1.154700538

Solving
2 + x = 1.154700538

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = 1.154700538 + -2

Combine like terms: 2 + -2 = 0
0 + x = 1.154700538 + -2
x = 1.154700538 + -2

Combine like terms: 1.154700538 + -2 = -0.845299462
x = -0.845299462

Simplifying
x = -0.845299462


Subproblem 2
x + 2 = -1.154700538

Simplifying
x + 2 = -1.154700538

Reorder the terms:
2 + x = -1.154700538

Solving
2 + x = -1.154700538

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = -1.154700538 + -2

Combine like terms: 2 + -2 = 0
0 + x = -1.154700538 + -2
x = -1.154700538 + -2

Combine like terms: -1.154700538 + -2 = -3.154700538
x = -3.154700538

Simplifying
x = -3.154700538


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.845299462, -3.154700538}
Solution
x = {-0.845299462, -3.154700538}

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