3r^2-12r-5=0

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Solution for 3r^2-12r-5=0 equation:


Simplifying
3r2 + -12r + -5 = 0

Reorder the terms:
-5 + -12r + 3r2 = 0

Solving
-5 + -12r + 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'.
-1.666666667 + -4r + r2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -4r + 1.666666667 + r2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -4r + r2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -4r + r2 = 0 + 1.666666667
-4r + r2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-4r + r2 = 1.666666667

The r term is -4r.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4r + 4 + r2 = 1.666666667 + 4

Reorder the terms:
4 + -4r + r2 = 1.666666667 + 4

Combine like terms: 1.666666667 + 4 = 5.666666667
4 + -4r + r2 = 5.666666667

Factor a perfect square on the left side:
(r + -2)(r + -2) = 5.666666667

Calculate the square root of the right side: 2.380476143

Break this problem into two subproblems by setting 
(r + -2) equal to 2.380476143 and -2.380476143.

Subproblem 1

r + -2 = 2.380476143 Simplifying r + -2 = 2.380476143 Reorder the terms: -2 + r = 2.380476143 Solving -2 + r = 2.380476143 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + r = 2.380476143 + 2 Combine like terms: -2 + 2 = 0 0 + r = 2.380476143 + 2 r = 2.380476143 + 2 Combine like terms: 2.380476143 + 2 = 4.380476143 r = 4.380476143 Simplifying r = 4.380476143

Subproblem 2

r + -2 = -2.380476143 Simplifying r + -2 = -2.380476143 Reorder the terms: -2 + r = -2.380476143 Solving -2 + r = -2.380476143 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + r = -2.380476143 + 2 Combine like terms: -2 + 2 = 0 0 + r = -2.380476143 + 2 r = -2.380476143 + 2 Combine like terms: -2.380476143 + 2 = -0.380476143 r = -0.380476143 Simplifying r = -0.380476143

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.380476143, -0.380476143}

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