3r^2-19r-12=0

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Solution for 3r^2-19r-12=0 equation: 


Simplifying
3r2 + -19r + -12 = 0

Reorder the terms:
-12 + -19r + 3r2 = 0

Solving
-12 + -19r + 3r2 = 0

Solving for variable 'r'.

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

Divide each side by '3'.
-4 + -6.333333333r + r2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -6.333333333r + 4 + r2 = 0 + 4

Reorder the terms:
-4 + 4 + -6.333333333r + r2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -6.333333333r + r2 = 0 + 4
-6.333333333r + r2 = 0 + 4

Combine like terms: 0 + 4 = 4
-6.333333333r + r2 = 4

The r term is -6.333333333r.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333r + 10.02777778 + r2 = 4 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333r + r2 = 4 + 10.02777778

Combine like terms: 4 + 10.02777778 = 14.02777778
10.02777778 + -6.333333333r + r2 = 14.02777778

Factor a perfect square on the left side:
(r + -3.166666667)(r + -3.166666667) = 14.02777778

Calculate the square root of the right side: 3.745367509

Break this problem into two subproblems by setting 
(r + -3.166666667) equal to 3.745367509 and -3.745367509.


Subproblem 1
r + -3.166666667 = 3.745367509

Simplifying
r + -3.166666667 = 3.745367509

Reorder the terms:
-3.166666667 + r = 3.745367509

Solving
-3.166666667 + r = 3.745367509

Solving for variable 'r'.

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

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + r = 3.745367509 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + r = 3.745367509 + 3.166666667
r = 3.745367509 + 3.166666667

Combine like terms: 3.745367509 + 3.166666667 = 6.912034176
r = 6.912034176

Simplifying
r = 6.912034176


Subproblem 2
r + -3.166666667 = -3.745367509

Simplifying
r + -3.166666667 = -3.745367509

Reorder the terms:
-3.166666667 + r = -3.745367509

Solving
-3.166666667 + r = -3.745367509

Solving for variable 'r'.

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

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + r = -3.745367509 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + r = -3.745367509 + 3.166666667
r = -3.745367509 + 3.166666667

Combine like terms: -3.745367509 + 3.166666667 = -0.578700842
r = -0.578700842

Simplifying
r = -0.578700842


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {6.912034176, -0.578700842}

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