3r^2-13r-4=0

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Solution for 3r^2-13r-4=0 equation: 


Simplifying
3r2 + -13r + -4 = 0

Reorder the terms:
-4 + -13r + 3r2 = 0

Solving
-4 + -13r + 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'.
-1.333333333 + -4.333333333r + r2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -4.333333333r + 1.333333333 + r2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -4.333333333r + r2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -4.333333333r + r2 = 0 + 1.333333333
-4.333333333r + r2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-4.333333333r + r2 = 1.333333333

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 = 1.333333333 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333r + r2 = 1.333333333 + 4.694444446

Combine like terms: 1.333333333 + 4.694444446 = 6.027777779
4.694444446 + -4.333333333r + r2 = 6.027777779

Factor a perfect square on the left side:
(r + -2.166666667)(r + -2.166666667) = 6.027777779

Calculate the square root of the right side: 2.455153311

Break this problem into two subproblems by setting 
(r + -2.166666667) equal to 2.455153311 and -2.455153311.


Subproblem 1
r + -2.166666667 = 2.455153311

Simplifying
r + -2.166666667 = 2.455153311

Reorder the terms:
-2.166666667 + r = 2.455153311

Solving
-2.166666667 + r = 2.455153311

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 = 2.455153311 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + r = 2.455153311 + 2.166666667
r = 2.455153311 + 2.166666667

Combine like terms: 2.455153311 + 2.166666667 = 4.621819978
r = 4.621819978

Simplifying
r = 4.621819978


Subproblem 2
r + -2.166666667 = -2.455153311

Simplifying
r + -2.166666667 = -2.455153311

Reorder the terms:
-2.166666667 + r = -2.455153311

Solving
-2.166666667 + r = -2.455153311

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 = -2.455153311 + 2.166666667

Combine like terms: -2.166666667 + 2.166666667 = 0.000000000
0.000000000 + r = -2.455153311 + 2.166666667
r = -2.455153311 + 2.166666667

Combine like terms: -2.455153311 + 2.166666667 = -0.288486644
r = -0.288486644

Simplifying
r = -0.288486644


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.621819978, -0.288486644}

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