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