-10+13r+3r^2=10

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Solution for -10+13r+3r^2=10 equation: 


Simplifying
-10 + 13r + 3r2 = 10

Solving
-10 + 13r + 3r2 = 10

Solving for variable 'r'.

Reorder the terms:
-10 + -10 + 13r + 3r2 = 10 + -10

Combine like terms: -10 + -10 = -20
-20 + 13r + 3r2 = 10 + -10

Combine like terms: 10 + -10 = 0
-20 + 13r + 3r2 = 0

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

Divide each side by '3'.
-6.666666667 + 4.333333333r + r2 = 0

Move the constant term to the right:

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

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

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

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

The r term is 4.333333333r.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333r + 4.694444446 + r2 = 6.666666667 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333r + r2 = 6.666666667 + 4.694444446

Combine like terms: 6.666666667 + 4.694444446 = 11.361111113
4.694444446 + 4.333333333r + r2 = 11.361111113

Factor a perfect square on the left side:
(r + 2.166666667)(r + 2.166666667) = 11.361111113

Calculate the square root of the right side: 3.370624736

Break this problem into two subproblems by setting 
(r + 2.166666667) equal to 3.370624736 and -3.370624736.


Subproblem 1
r + 2.166666667 = 3.370624736

Simplifying
r + 2.166666667 = 3.370624736

Reorder the terms:
2.166666667 + r = 3.370624736

Solving
2.166666667 + r = 3.370624736

Solving for variable 'r'.

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

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + r = 3.370624736 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + r = 3.370624736 + -2.166666667
r = 3.370624736 + -2.166666667

Combine like terms: 3.370624736 + -2.166666667 = 1.203958069
r = 1.203958069

Simplifying
r = 1.203958069


Subproblem 2
r + 2.166666667 = -3.370624736

Simplifying
r + 2.166666667 = -3.370624736

Reorder the terms:
2.166666667 + r = -3.370624736

Solving
2.166666667 + r = -3.370624736

Solving for variable 'r'.

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

Add '-2.166666667' to each side of the equation.
2.166666667 + -2.166666667 + r = -3.370624736 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + r = -3.370624736 + -2.166666667
r = -3.370624736 + -2.166666667

Combine like terms: -3.370624736 + -2.166666667 = -5.537291403
r = -5.537291403

Simplifying
r = -5.537291403


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.203958069, -5.537291403}

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