6r^2-12r-9=0

Solution for 6r^2-12r-9=0 equation:

Simplifying
6r2 + -12r + -9 = 0

Reorder the terms:
-9 + -12r + 6r2 = 0

Solving
-9 + -12r + 6r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-3 + -4r + 2r2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-3 + -4r + 2r2)' equal to zero and attempt to solve:

Simplifying
-3 + -4r + 2r2 = 0

Solving
-3 + -4r + 2r2 = 0

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

Divide each side by '2'.
-1.5 + -2r + r2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -2r + 1.5 + r2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -2r + r2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -2r + r2 = 0 + 1.5
-2r + r2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-2r + r2 = 1.5

The r term is -2r.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2r + 1 + r2 = 1.5 + 1

Reorder the terms:
1 + -2r + r2 = 1.5 + 1

Combine like terms: 1.5 + 1 = 2.5
1 + -2r + r2 = 2.5

Factor a perfect square on the left side:
(r + -1)(r + -1) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(r + -1) equal to 1.58113883 and -1.58113883.

Subproblem 1
r + -1 = 1.58113883

Simplifying
r + -1 = 1.58113883

Reorder the terms:
-1 + r = 1.58113883

Solving
-1 + r = 1.58113883

Solving for variable 'r'.

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

Add '1' to each side of the equation.
-1 + 1 + r = 1.58113883 + 1

Combine like terms: -1 + 1 = 0
0 + r = 1.58113883 + 1
r = 1.58113883 + 1

Combine like terms: 1.58113883 + 1 = 2.58113883
r = 2.58113883

Simplifying
r = 2.58113883

Subproblem 2
r + -1 = -1.58113883

Simplifying
r + -1 = -1.58113883

Reorder the terms:
-1 + r = -1.58113883

Solving
-1 + r = -1.58113883

Solving for variable 'r'.

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

Add '1' to each side of the equation.
-1 + 1 + r = -1.58113883 + 1

Combine like terms: -1 + 1 = 0
0 + r = -1.58113883 + 1
r = -1.58113883 + 1

Combine like terms: -1.58113883 + 1 = -0.58113883
r = -0.58113883

Simplifying
r = -0.58113883

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {2.58113883, -0.58113883}
Solution
r = {2.58113883, -0.58113883}