3r^2+20r+13=0

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Solution for 3r^2+20r+13=0 equation: 


Simplifying
3r2 + 20r + 13 = 0

Reorder the terms:
13 + 20r + 3r2 = 0

Solving
13 + 20r + 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'.
4.333333333 + 6.666666667r + r2 = 0

Move the constant term to the right:

Add '-4.333333333' to each side of the equation.
4.333333333 + 6.666666667r + -4.333333333 + r2 = 0 + -4.333333333

Reorder the terms:
4.333333333 + -4.333333333 + 6.666666667r + r2 = 0 + -4.333333333

Combine like terms: 4.333333333 + -4.333333333 = 0.000000000
0.000000000 + 6.666666667r + r2 = 0 + -4.333333333
6.666666667r + r2 = 0 + -4.333333333

Combine like terms: 0 + -4.333333333 = -4.333333333
6.666666667r + r2 = -4.333333333

The r term is 6.666666667r.  Take half its coefficient (3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
6.666666667r + 11.11111112 + r2 = -4.333333333 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667r + r2 = -4.333333333 + 11.11111112

Combine like terms: -4.333333333 + 11.11111112 = 6.777777787
11.11111112 + 6.666666667r + r2 = 6.777777787

Factor a perfect square on the left side:
(r + 3.333333334)(r + 3.333333334) = 6.777777787

Calculate the square root of the right side: 2.60341656

Break this problem into two subproblems by setting 
(r + 3.333333334) equal to 2.60341656 and -2.60341656.


Subproblem 1
r + 3.333333334 = 2.60341656

Simplifying
r + 3.333333334 = 2.60341656

Reorder the terms:
3.333333334 + r = 2.60341656

Solving
3.333333334 + r = 2.60341656

Solving for variable 'r'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + r = 2.60341656 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + r = 2.60341656 + -3.333333334
r = 2.60341656 + -3.333333334

Combine like terms: 2.60341656 + -3.333333334 = -0.729916774
r = -0.729916774

Simplifying
r = -0.729916774


Subproblem 2
r + 3.333333334 = -2.60341656

Simplifying
r + 3.333333334 = -2.60341656

Reorder the terms:
3.333333334 + r = -2.60341656

Solving
3.333333334 + r = -2.60341656

Solving for variable 'r'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + r = -2.60341656 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + r = -2.60341656 + -3.333333334
r = -2.60341656 + -3.333333334

Combine like terms: -2.60341656 + -3.333333334 = -5.936749894
r = -5.936749894

Simplifying
r = -5.936749894


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-0.729916774, -5.936749894}

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