6r^2+31r+8=0

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Solution for 6r^2+31r+8=0 equation: 


Simplifying
6r2 + 31r + 8 = 0

Reorder the terms:
8 + 31r + 6r2 = 0

Solving
8 + 31r + 6r2 = 0

Solving for variable 'r'.

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

Divide each side by '6'.
1.333333333 + 5.166666667r + r2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + 5.166666667r + -1.333333333 + r2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + 5.166666667r + r2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + 5.166666667r + r2 = 0 + -1.333333333
5.166666667r + r2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
5.166666667r + r2 = -1.333333333

The r term is 5.166666667r.  Take half its coefficient (2.583333334).
Square it (6.673611115) and add it to both sides.

Add '6.673611115' to each side of the equation.
5.166666667r + 6.673611115 + r2 = -1.333333333 + 6.673611115

Reorder the terms:
6.673611115 + 5.166666667r + r2 = -1.333333333 + 6.673611115

Combine like terms: -1.333333333 + 6.673611115 = 5.340277782
6.673611115 + 5.166666667r + r2 = 5.340277782

Factor a perfect square on the left side:
(r + 2.583333334)(r + 2.583333334) = 5.340277782

Calculate the square root of the right side: 2.310904105

Break this problem into two subproblems by setting 
(r + 2.583333334) equal to 2.310904105 and -2.310904105.


Subproblem 1
r + 2.583333334 = 2.310904105

Simplifying
r + 2.583333334 = 2.310904105

Reorder the terms:
2.583333334 + r = 2.310904105

Solving
2.583333334 + r = 2.310904105

Solving for variable 'r'.

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

Add '-2.583333334' to each side of the equation.
2.583333334 + -2.583333334 + r = 2.310904105 + -2.583333334

Combine like terms: 2.583333334 + -2.583333334 = 0.000000000
0.000000000 + r = 2.310904105 + -2.583333334
r = 2.310904105 + -2.583333334

Combine like terms: 2.310904105 + -2.583333334 = -0.272429229
r = -0.272429229

Simplifying
r = -0.272429229


Subproblem 2
r + 2.583333334 = -2.310904105

Simplifying
r + 2.583333334 = -2.310904105

Reorder the terms:
2.583333334 + r = -2.310904105

Solving
2.583333334 + r = -2.310904105

Solving for variable 'r'.

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

Add '-2.583333334' to each side of the equation.
2.583333334 + -2.583333334 + r = -2.310904105 + -2.583333334

Combine like terms: 2.583333334 + -2.583333334 = 0.000000000
0.000000000 + r = -2.310904105 + -2.583333334
r = -2.310904105 + -2.583333334

Combine like terms: -2.310904105 + -2.583333334 = -4.894237439
r = -4.894237439

Simplifying
r = -4.894237439


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-0.272429229, -4.894237439}

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