3r^2+1r-9=0

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Solution for 3r^2+1r-9=0 equation: 


Simplifying
3r2 + 1r + -9 = 0

Reorder the terms:
-9 + 1r + 3r2 = 0

Solving
-9 + 1r + 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'.
-3 + 0.3333333333r + r2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 0.3333333333r + 3 + r2 = 0 + 3

Reorder the terms:
-3 + 3 + 0.3333333333r + r2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 0.3333333333r + r2 = 0 + 3
0.3333333333r + r2 = 0 + 3

Combine like terms: 0 + 3 = 3
0.3333333333r + r2 = 3

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 = 3 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333r + r2 = 3 + 0.02777777779

Combine like terms: 3 + 0.02777777779 = 3.02777777779
0.02777777779 + 0.3333333333r + r2 = 3.02777777779

Factor a perfect square on the left side:
(r + 0.1666666667)(r + 0.1666666667) = 3.02777777779

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
(r + 0.1666666667) equal to 1.740051085 and -1.740051085.


Subproblem 1
r + 0.1666666667 = 1.740051085

Simplifying
r + 0.1666666667 = 1.740051085

Reorder the terms:
0.1666666667 + r = 1.740051085

Solving
0.1666666667 + r = 1.740051085

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 = 1.740051085 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + r = 1.740051085 + -0.1666666667
r = 1.740051085 + -0.1666666667

Combine like terms: 1.740051085 + -0.1666666667 = 1.5733844183
r = 1.5733844183

Simplifying
r = 1.5733844183


Subproblem 2
r + 0.1666666667 = -1.740051085

Simplifying
r + 0.1666666667 = -1.740051085

Reorder the terms:
0.1666666667 + r = -1.740051085

Solving
0.1666666667 + r = -1.740051085

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 = -1.740051085 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + r = -1.740051085 + -0.1666666667
r = -1.740051085 + -0.1666666667

Combine like terms: -1.740051085 + -0.1666666667 = -1.9067177517
r = -1.9067177517

Simplifying
r = -1.9067177517


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.5733844183, -1.9067177517}

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