6X^2+34X+16=0

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Solution for 6X^2+34X+16=0 equation: 


Simplifying
6X2 + 34X + 16 = 0

Reorder the terms:
16 + 34X + 6X2 = 0

Solving
16 + 34X + 6X2 = 0

Solving for variable 'X'.

Factor out the Greatest Common Factor (GCF), '2'.
2(8 + 17X + 3X2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(8 + 17X + 3X2)' equal to zero and attempt to solve:

Simplifying
8 + 17X + 3X2 = 0

Solving
8 + 17X + 3X2 = 0

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

Divide each side by '3'.
2.666666667 + 5.666666667X + X2 = 0

Move the constant term to the right:

Add '-2.666666667' to each side of the equation.
2.666666667 + 5.666666667X + -2.666666667 + X2 = 0 + -2.666666667

Reorder the terms:
2.666666667 + -2.666666667 + 5.666666667X + X2 = 0 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + 5.666666667X + X2 = 0 + -2.666666667
5.666666667X + X2 = 0 + -2.666666667

Combine like terms: 0 + -2.666666667 = -2.666666667
5.666666667X + X2 = -2.666666667

The X term is 5.666666667X.  Take half its coefficient (2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
5.666666667X + 8.027777782 + X2 = -2.666666667 + 8.027777782

Reorder the terms:
8.027777782 + 5.666666667X + X2 = -2.666666667 + 8.027777782

Combine like terms: -2.666666667 + 8.027777782 = 5.361111115
8.027777782 + 5.666666667X + X2 = 5.361111115

Factor a perfect square on the left side:
(X + 2.833333334)(X + 2.833333334) = 5.361111115

Calculate the square root of the right side: 2.315407332

Break this problem into two subproblems by setting 
(X + 2.833333334) equal to 2.315407332 and -2.315407332.


Subproblem 1
X + 2.833333334 = 2.315407332

Simplifying
X + 2.833333334 = 2.315407332

Reorder the terms:
2.833333334 + X = 2.315407332

Solving
2.833333334 + X = 2.315407332

Solving for variable 'X'.

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

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + X = 2.315407332 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + X = 2.315407332 + -2.833333334
X = 2.315407332 + -2.833333334

Combine like terms: 2.315407332 + -2.833333334 = -0.517926002
X = -0.517926002

Simplifying
X = -0.517926002


Subproblem 2
X + 2.833333334 = -2.315407332

Simplifying
X + 2.833333334 = -2.315407332

Reorder the terms:
2.833333334 + X = -2.315407332

Solving
2.833333334 + X = -2.315407332

Solving for variable 'X'.

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

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + X = -2.315407332 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + X = -2.315407332 + -2.833333334
X = -2.315407332 + -2.833333334

Combine like terms: -2.315407332 + -2.833333334 = -5.148740666
X = -5.148740666

Simplifying
X = -5.148740666


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {-0.517926002, -5.148740666}
Solution
X = {-0.517926002, -5.148740666}

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