12r^2=19+3r

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Solution for 12r^2=19+3r equation:


Simplifying
12r2 = 19 + 3r

Solving
12r2 = 19 + 3r

Solving for variable 'r'.

Reorder the terms:
-19 + -3r + 12r2 = 19 + 3r + -19 + -3r

Reorder the terms:
-19 + -3r + 12r2 = 19 + -19 + 3r + -3r

Combine like terms: 19 + -19 = 0
-19 + -3r + 12r2 = 0 + 3r + -3r
-19 + -3r + 12r2 = 3r + -3r

Combine like terms: 3r + -3r = 0
-19 + -3r + 12r2 = 0

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

Divide each side by '12'.
-1.583333333 + -0.25r + r2 = 0

Move the constant term to the right:

Add '1.583333333' to each side of the equation.
-1.583333333 + -0.25r + 1.583333333 + r2 = 0 + 1.583333333

Reorder the terms:
-1.583333333 + 1.583333333 + -0.25r + r2 = 0 + 1.583333333

Combine like terms: -1.583333333 + 1.583333333 = 0.000000000
0.000000000 + -0.25r + r2 = 0 + 1.583333333
-0.25r + r2 = 0 + 1.583333333

Combine like terms: 0 + 1.583333333 = 1.583333333
-0.25r + r2 = 1.583333333

The r term is -0.25r.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25r + 0.015625 + r2 = 1.583333333 + 0.015625

Reorder the terms:
0.015625 + -0.25r + r2 = 1.583333333 + 0.015625

Combine like terms: 1.583333333 + 0.015625 = 1.598958333
0.015625 + -0.25r + r2 = 1.598958333

Factor a perfect square on the left side:
(r + -0.125)(r + -0.125) = 1.598958333

Calculate the square root of the right side: 1.264499242

Break this problem into two subproblems by setting 
(r + -0.125) equal to 1.264499242 and -1.264499242.

Subproblem 1

r + -0.125 = 1.264499242 Simplifying r + -0.125 = 1.264499242 Reorder the terms: -0.125 + r = 1.264499242 Solving -0.125 + r = 1.264499242 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + r = 1.264499242 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + r = 1.264499242 + 0.125 r = 1.264499242 + 0.125 Combine like terms: 1.264499242 + 0.125 = 1.389499242 r = 1.389499242 Simplifying r = 1.389499242

Subproblem 2

r + -0.125 = -1.264499242 Simplifying r + -0.125 = -1.264499242 Reorder the terms: -0.125 + r = -1.264499242 Solving -0.125 + r = -1.264499242 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + r = -1.264499242 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + r = -1.264499242 + 0.125 r = -1.264499242 + 0.125 Combine like terms: -1.264499242 + 0.125 = -1.139499242 r = -1.139499242 Simplifying r = -1.139499242

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.389499242, -1.139499242}

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