6x^2+29.5x+16=0

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Solution for 6x^2+29.5x+16=0 equation: 


Simplifying
6x2 + 29.5x + 16 = 0

Reorder the terms:
16 + 29.5x + 6x2 = 0

Solving
16 + 29.5x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
2.666666667 + 4.916666667x + x2 = 0

Move the constant term to the right:

Add '-2.666666667' to each side of the equation.
2.666666667 + 4.916666667x + -2.666666667 + x2 = 0 + -2.666666667

Reorder the terms:
2.666666667 + -2.666666667 + 4.916666667x + x2 = 0 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + 4.916666667x + x2 = 0 + -2.666666667
4.916666667x + x2 = 0 + -2.666666667

Combine like terms: 0 + -2.666666667 = -2.666666667
4.916666667x + x2 = -2.666666667

The x term is 4.916666667x.  Take half its coefficient (2.458333334).
Square it (6.043402781) and add it to both sides.

Add '6.043402781' to each side of the equation.
4.916666667x + 6.043402781 + x2 = -2.666666667 + 6.043402781

Reorder the terms:
6.043402781 + 4.916666667x + x2 = -2.666666667 + 6.043402781

Combine like terms: -2.666666667 + 6.043402781 = 3.376736114
6.043402781 + 4.916666667x + x2 = 3.376736114

Factor a perfect square on the left side:
(x + 2.458333334)(x + 2.458333334) = 3.376736114

Calculate the square root of the right side: 1.837589757

Break this problem into two subproblems by setting 
(x + 2.458333334) equal to 1.837589757 and -1.837589757.


Subproblem 1
x + 2.458333334 = 1.837589757

Simplifying
x + 2.458333334 = 1.837589757

Reorder the terms:
2.458333334 + x = 1.837589757

Solving
2.458333334 + x = 1.837589757

Solving for variable 'x'.

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

Add '-2.458333334' to each side of the equation.
2.458333334 + -2.458333334 + x = 1.837589757 + -2.458333334

Combine like terms: 2.458333334 + -2.458333334 = 0.000000000
0.000000000 + x = 1.837589757 + -2.458333334
x = 1.837589757 + -2.458333334

Combine like terms: 1.837589757 + -2.458333334 = -0.620743577
x = -0.620743577

Simplifying
x = -0.620743577


Subproblem 2
x + 2.458333334 = -1.837589757

Simplifying
x + 2.458333334 = -1.837589757

Reorder the terms:
2.458333334 + x = -1.837589757

Solving
2.458333334 + x = -1.837589757

Solving for variable 'x'.

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

Add '-2.458333334' to each side of the equation.
2.458333334 + -2.458333334 + x = -1.837589757 + -2.458333334

Combine like terms: 2.458333334 + -2.458333334 = 0.000000000
0.000000000 + x = -1.837589757 + -2.458333334
x = -1.837589757 + -2.458333334

Combine like terms: -1.837589757 + -2.458333334 = -4.295923091
x = -4.295923091

Simplifying
x = -4.295923091


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.620743577, -4.295923091}

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