6r^2-2r-3=0

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Solution for 6r^2-2r-3=0 equation: 


Simplifying
6r2 + -2r + -3 = 0

Reorder the terms:
-3 + -2r + 6r2 = 0

Solving
-3 + -2r + 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'.
-0.5 + -0.3333333333r + r2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.3333333333r + 0.5 + r2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.3333333333r + r2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.3333333333r + r2 = 0 + 0.5
-0.3333333333r + r2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.3333333333r + r2 = 0.5

The r term is -0.3333333333r.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333r + 0.02777777779 + r2 = 0.5 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333r + r2 = 0.5 + 0.02777777779

Combine like terms: 0.5 + 0.02777777779 = 0.52777777779
0.02777777779 + -0.3333333333r + r2 = 0.52777777779

Factor a perfect square on the left side:
(r + -0.1666666667)(r + -0.1666666667) = 0.52777777779

Calculate the square root of the right side: 0.726483157

Break this problem into two subproblems by setting 
(r + -0.1666666667) equal to 0.726483157 and -0.726483157.


Subproblem 1
r + -0.1666666667 = 0.726483157

Simplifying
r + -0.1666666667 = 0.726483157

Reorder the terms:
-0.1666666667 + r = 0.726483157

Solving
-0.1666666667 + r = 0.726483157

Solving for variable 'r'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + r = 0.726483157 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + r = 0.726483157 + 0.1666666667
r = 0.726483157 + 0.1666666667

Combine like terms: 0.726483157 + 0.1666666667 = 0.8931498237
r = 0.8931498237

Simplifying
r = 0.8931498237


Subproblem 2
r + -0.1666666667 = -0.726483157

Simplifying
r + -0.1666666667 = -0.726483157

Reorder the terms:
-0.1666666667 + r = -0.726483157

Solving
-0.1666666667 + r = -0.726483157

Solving for variable 'r'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + r = -0.726483157 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + r = -0.726483157 + 0.1666666667
r = -0.726483157 + 0.1666666667

Combine like terms: -0.726483157 + 0.1666666667 = -0.5598164903
r = -0.5598164903

Simplifying
r = -0.5598164903


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {0.8931498237, -0.5598164903}

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