6r^2-24r+22=0

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Solution for 6r^2-24r+22=0 equation: 


Simplifying
6r2 + -24r + 22 = 0

Reorder the terms:
22 + -24r + 6r2 = 0

Solving
22 + -24r + 6r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '2'.
2(11 + -12r + 3r2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(11 + -12r + 3r2)' equal to zero and attempt to solve:

Simplifying
11 + -12r + 3r2 = 0

Solving
11 + -12r + 3r2 = 0

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

Divide each side by '3'.
3.666666667 + -4r + r2 = 0

Move the constant term to the right:

Add '-3.666666667' to each side of the equation.
3.666666667 + -4r + -3.666666667 + r2 = 0 + -3.666666667

Reorder the terms:
3.666666667 + -3.666666667 + -4r + r2 = 0 + -3.666666667

Combine like terms: 3.666666667 + -3.666666667 = 0.000000000
0.000000000 + -4r + r2 = 0 + -3.666666667
-4r + r2 = 0 + -3.666666667

Combine like terms: 0 + -3.666666667 = -3.666666667
-4r + r2 = -3.666666667

The r term is -4r.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4r + 4 + r2 = -3.666666667 + 4

Reorder the terms:
4 + -4r + r2 = -3.666666667 + 4

Combine like terms: -3.666666667 + 4 = 0.333333333
4 + -4r + r2 = 0.333333333

Factor a perfect square on the left side:
(r + -2)(r + -2) = 0.333333333

Calculate the square root of the right side: 0.577350269

Break this problem into two subproblems by setting 
(r + -2) equal to 0.577350269 and -0.577350269.


Subproblem 1
r + -2 = 0.577350269

Simplifying
r + -2 = 0.577350269

Reorder the terms:
-2 + r = 0.577350269

Solving
-2 + r = 0.577350269

Solving for variable 'r'.

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

Add '2' to each side of the equation.
-2 + 2 + r = 0.577350269 + 2

Combine like terms: -2 + 2 = 0
0 + r = 0.577350269 + 2
r = 0.577350269 + 2

Combine like terms: 0.577350269 + 2 = 2.577350269
r = 2.577350269

Simplifying
r = 2.577350269


Subproblem 2
r + -2 = -0.577350269

Simplifying
r + -2 = -0.577350269

Reorder the terms:
-2 + r = -0.577350269

Solving
-2 + r = -0.577350269

Solving for variable 'r'.

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

Add '2' to each side of the equation.
-2 + 2 + r = -0.577350269 + 2

Combine like terms: -2 + 2 = 0
0 + r = -0.577350269 + 2
r = -0.577350269 + 2

Combine like terms: -0.577350269 + 2 = 1.422649731
r = 1.422649731

Simplifying
r = 1.422649731


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {2.577350269, 1.422649731}
Solution
r = {2.577350269, 1.422649731}

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