3r^2-r-15=0

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Solution for 3r^2-r-15=0 equation: 


Simplifying
3r2 + -1r + -15 = 0

Reorder the terms:
-15 + -1r + 3r2 = 0

Solving
-15 + -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'.
-5 + -0.3333333333r + r2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -0.3333333333r + 5 + r2 = 0 + 5

Reorder the terms:
-5 + 5 + -0.3333333333r + r2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -0.3333333333r + r2 = 0 + 5
-0.3333333333r + r2 = 0 + 5

Combine like terms: 0 + 5 = 5
-0.3333333333r + r2 = 5

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 = 5 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333r + r2 = 5 + 0.02777777779

Combine like terms: 5 + 0.02777777779 = 5.02777777779
0.02777777779 + -0.3333333333r + r2 = 5.02777777779

Factor a perfect square on the left side:
(r + -0.1666666667)(r + -0.1666666667) = 5.02777777779

Calculate the square root of the right side: 2.242270675

Break this problem into two subproblems by setting 
(r + -0.1666666667) equal to 2.242270675 and -2.242270675.


Subproblem 1
r + -0.1666666667 = 2.242270675

Simplifying
r + -0.1666666667 = 2.242270675

Reorder the terms:
-0.1666666667 + r = 2.242270675

Solving
-0.1666666667 + r = 2.242270675

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 = 2.242270675 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + r = 2.242270675 + 0.1666666667
r = 2.242270675 + 0.1666666667

Combine like terms: 2.242270675 + 0.1666666667 = 2.4089373417
r = 2.4089373417

Simplifying
r = 2.4089373417


Subproblem 2
r + -0.1666666667 = -2.242270675

Simplifying
r + -0.1666666667 = -2.242270675

Reorder the terms:
-0.1666666667 + r = -2.242270675

Solving
-0.1666666667 + r = -2.242270675

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 = -2.242270675 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + r = -2.242270675 + 0.1666666667
r = -2.242270675 + 0.1666666667

Combine like terms: -2.242270675 + 0.1666666667 = -2.0756040083
r = -2.0756040083

Simplifying
r = -2.0756040083


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {2.4089373417, -2.0756040083}

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