6r^2-19r-4=0

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Solution for 6r^2-19r-4=0 equation: 


Simplifying
6r2 + -19r + -4 = 0

Reorder the terms:
-4 + -19r + 6r2 = 0

Solving
-4 + -19r + 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'.
-0.6666666667 + -3.166666667r + r2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -3.166666667r + 0.6666666667 + r2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -3.166666667r + r2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -3.166666667r + r2 = 0 + 0.6666666667
-3.166666667r + r2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-3.166666667r + r2 = 0.6666666667

The r term is -3.166666667r.  Take half its coefficient (-1.583333334).
Square it (2.506944447) and add it to both sides.

Add '2.506944447' to each side of the equation.
-3.166666667r + 2.506944447 + r2 = 0.6666666667 + 2.506944447

Reorder the terms:
2.506944447 + -3.166666667r + r2 = 0.6666666667 + 2.506944447

Combine like terms: 0.6666666667 + 2.506944447 = 3.1736111137
2.506944447 + -3.166666667r + r2 = 3.1736111137

Factor a perfect square on the left side:
(r + -1.583333334)(r + -1.583333334) = 3.1736111137

Calculate the square root of the right side: 1.781463195

Break this problem into two subproblems by setting 
(r + -1.583333334) equal to 1.781463195 and -1.781463195.


Subproblem 1
r + -1.583333334 = 1.781463195

Simplifying
r + -1.583333334 = 1.781463195

Reorder the terms:
-1.583333334 + r = 1.781463195

Solving
-1.583333334 + r = 1.781463195

Solving for variable 'r'.

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

Add '1.583333334' to each side of the equation.
-1.583333334 + 1.583333334 + r = 1.781463195 + 1.583333334

Combine like terms: -1.583333334 + 1.583333334 = 0.000000000
0.000000000 + r = 1.781463195 + 1.583333334
r = 1.781463195 + 1.583333334

Combine like terms: 1.781463195 + 1.583333334 = 3.364796529
r = 3.364796529

Simplifying
r = 3.364796529


Subproblem 2
r + -1.583333334 = -1.781463195

Simplifying
r + -1.583333334 = -1.781463195

Reorder the terms:
-1.583333334 + r = -1.781463195

Solving
-1.583333334 + r = -1.781463195

Solving for variable 'r'.

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

Add '1.583333334' to each side of the equation.
-1.583333334 + 1.583333334 + r = -1.781463195 + 1.583333334

Combine like terms: -1.583333334 + 1.583333334 = 0.000000000
0.000000000 + r = -1.781463195 + 1.583333334
r = -1.781463195 + 1.583333334

Combine like terms: -1.781463195 + 1.583333334 = -0.198129861
r = -0.198129861

Simplifying
r = -0.198129861


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {3.364796529, -0.198129861}

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