3r^2-31r-105=0

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Solution for 3r^2-31r-105=0 equation: 


Simplifying
3r2 + -31r + -105 = 0

Reorder the terms:
-105 + -31r + 3r2 = 0

Solving
-105 + -31r + 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'.
-35 + -10.33333333r + r2 = 0

Move the constant term to the right:

Add '35' to each side of the equation.
-35 + -10.33333333r + 35 + r2 = 0 + 35

Reorder the terms:
-35 + 35 + -10.33333333r + r2 = 0 + 35

Combine like terms: -35 + 35 = 0
0 + -10.33333333r + r2 = 0 + 35
-10.33333333r + r2 = 0 + 35

Combine like terms: 0 + 35 = 35
-10.33333333r + r2 = 35

The r term is -10.33333333r.  Take half its coefficient (-5.166666665).
Square it (26.69444443) and add it to both sides.

Add '26.69444443' to each side of the equation.
-10.33333333r + 26.69444443 + r2 = 35 + 26.69444443

Reorder the terms:
26.69444443 + -10.33333333r + r2 = 35 + 26.69444443

Combine like terms: 35 + 26.69444443 = 61.69444443
26.69444443 + -10.33333333r + r2 = 61.69444443

Factor a perfect square on the left side:
(r + -5.166666665)(r + -5.166666665) = 61.69444443

Calculate the square root of the right side: 7.854581111

Break this problem into two subproblems by setting 
(r + -5.166666665) equal to 7.854581111 and -7.854581111.


Subproblem 1
r + -5.166666665 = 7.854581111

Simplifying
r + -5.166666665 = 7.854581111

Reorder the terms:
-5.166666665 + r = 7.854581111

Solving
-5.166666665 + r = 7.854581111

Solving for variable 'r'.

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

Add '5.166666665' to each side of the equation.
-5.166666665 + 5.166666665 + r = 7.854581111 + 5.166666665

Combine like terms: -5.166666665 + 5.166666665 = 0.000000000
0.000000000 + r = 7.854581111 + 5.166666665
r = 7.854581111 + 5.166666665

Combine like terms: 7.854581111 + 5.166666665 = 13.021247776
r = 13.021247776

Simplifying
r = 13.021247776


Subproblem 2
r + -5.166666665 = -7.854581111

Simplifying
r + -5.166666665 = -7.854581111

Reorder the terms:
-5.166666665 + r = -7.854581111

Solving
-5.166666665 + r = -7.854581111

Solving for variable 'r'.

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

Add '5.166666665' to each side of the equation.
-5.166666665 + 5.166666665 + r = -7.854581111 + 5.166666665

Combine like terms: -5.166666665 + 5.166666665 = 0.000000000
0.000000000 + r = -7.854581111 + 5.166666665
r = -7.854581111 + 5.166666665

Combine like terms: -7.854581111 + 5.166666665 = -2.687914446
r = -2.687914446

Simplifying
r = -2.687914446


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {13.021247776, -2.687914446}

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