6x^2+49x+82=0

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Solution for 6x^2+49x+82=0 equation: 


Simplifying
6x2 + 49x + 82 = 0

Reorder the terms:
82 + 49x + 6x2 = 0

Solving
82 + 49x + 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'.
13.66666667 + 8.166666667x + x2 = 0

Move the constant term to the right:

Add '-13.66666667' to each side of the equation.
13.66666667 + 8.166666667x + -13.66666667 + x2 = 0 + -13.66666667

Reorder the terms:
13.66666667 + -13.66666667 + 8.166666667x + x2 = 0 + -13.66666667

Combine like terms: 13.66666667 + -13.66666667 = 0.00000000
0.00000000 + 8.166666667x + x2 = 0 + -13.66666667
8.166666667x + x2 = 0 + -13.66666667

Combine like terms: 0 + -13.66666667 = -13.66666667
8.166666667x + x2 = -13.66666667

The x term is 8.166666667x.  Take half its coefficient (4.083333334).
Square it (16.67361112) and add it to both sides.

Add '16.67361112' to each side of the equation.
8.166666667x + 16.67361112 + x2 = -13.66666667 + 16.67361112

Reorder the terms:
16.67361112 + 8.166666667x + x2 = -13.66666667 + 16.67361112

Combine like terms: -13.66666667 + 16.67361112 = 3.00694445
16.67361112 + 8.166666667x + x2 = 3.00694445

Factor a perfect square on the left side:
(x + 4.083333334)(x + 4.083333334) = 3.00694445

Calculate the square root of the right side: 1.734054339

Break this problem into two subproblems by setting 
(x + 4.083333334) equal to 1.734054339 and -1.734054339.


Subproblem 1
x + 4.083333334 = 1.734054339

Simplifying
x + 4.083333334 = 1.734054339

Reorder the terms:
4.083333334 + x = 1.734054339

Solving
4.083333334 + x = 1.734054339

Solving for variable 'x'.

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

Add '-4.083333334' to each side of the equation.
4.083333334 + -4.083333334 + x = 1.734054339 + -4.083333334

Combine like terms: 4.083333334 + -4.083333334 = 0.000000000
0.000000000 + x = 1.734054339 + -4.083333334
x = 1.734054339 + -4.083333334

Combine like terms: 1.734054339 + -4.083333334 = -2.349278995
x = -2.349278995

Simplifying
x = -2.349278995


Subproblem 2
x + 4.083333334 = -1.734054339

Simplifying
x + 4.083333334 = -1.734054339

Reorder the terms:
4.083333334 + x = -1.734054339

Solving
4.083333334 + x = -1.734054339

Solving for variable 'x'.

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

Add '-4.083333334' to each side of the equation.
4.083333334 + -4.083333334 + x = -1.734054339 + -4.083333334

Combine like terms: 4.083333334 + -4.083333334 = 0.000000000
0.000000000 + x = -1.734054339 + -4.083333334
x = -1.734054339 + -4.083333334

Combine like terms: -1.734054339 + -4.083333334 = -5.817387673
x = -5.817387673

Simplifying
x = -5.817387673


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.349278995, -5.817387673}

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