12r^2-13r-20=0

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Solution for 12r^2-13r-20=0 equation: 


Simplifying
12r2 + -13r + -20 = 0

Reorder the terms:
-20 + -13r + 12r2 = 0

Solving
-20 + -13r + 12r2 = 0

Solving for variable 'r'.

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

Divide each side by '12'.
-1.666666667 + -1.083333333r + r2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -1.083333333r + 1.666666667 + r2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -1.083333333r + r2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -1.083333333r + r2 = 0 + 1.666666667
-1.083333333r + r2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-1.083333333r + r2 = 1.666666667

The r term is -1.083333333r.  Take half its coefficient (-0.5416666665).
Square it (0.2934027776) and add it to both sides.

Add '0.2934027776' to each side of the equation.
-1.083333333r + 0.2934027776 + r2 = 1.666666667 + 0.2934027776

Reorder the terms:
0.2934027776 + -1.083333333r + r2 = 1.666666667 + 0.2934027776

Combine like terms: 1.666666667 + 0.2934027776 = 1.9600694446
0.2934027776 + -1.083333333r + r2 = 1.9600694446

Factor a perfect square on the left side:
(r + -0.5416666665)(r + -0.5416666665) = 1.9600694446

Calculate the square root of the right side: 1.400024801

Break this problem into two subproblems by setting 
(r + -0.5416666665) equal to 1.400024801 and -1.400024801.


Subproblem 1
r + -0.5416666665 = 1.400024801

Simplifying
r + -0.5416666665 = 1.400024801

Reorder the terms:
-0.5416666665 + r = 1.400024801

Solving
-0.5416666665 + r = 1.400024801

Solving for variable 'r'.

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

Add '0.5416666665' to each side of the equation.
-0.5416666665 + 0.5416666665 + r = 1.400024801 + 0.5416666665

Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000
0.0000000000 + r = 1.400024801 + 0.5416666665
r = 1.400024801 + 0.5416666665

Combine like terms: 1.400024801 + 0.5416666665 = 1.9416914675
r = 1.9416914675

Simplifying
r = 1.9416914675


Subproblem 2
r + -0.5416666665 = -1.400024801

Simplifying
r + -0.5416666665 = -1.400024801

Reorder the terms:
-0.5416666665 + r = -1.400024801

Solving
-0.5416666665 + r = -1.400024801

Solving for variable 'r'.

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

Add '0.5416666665' to each side of the equation.
-0.5416666665 + 0.5416666665 + r = -1.400024801 + 0.5416666665

Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000
0.0000000000 + r = -1.400024801 + 0.5416666665
r = -1.400024801 + 0.5416666665

Combine like terms: -1.400024801 + 0.5416666665 = -0.8583581345
r = -0.8583581345

Simplifying
r = -0.8583581345


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.9416914675, -0.8583581345}

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