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Simplifying 103.9r * r + 103.9r + -31.1 = 0 Multiply r * r 103.9r2 + 103.9r + -31.1 = 0 Reorder the terms: -31.1 + 103.9r + 103.9r2 = 0 Solving -31.1 + 103.9r + 103.9r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 103.9 the coefficient of the squared term: Divide each side by '103.9'. -0.2993262753 + r + r2 = 0 Move the constant term to the right: Add '0.2993262753' to each side of the equation. -0.2993262753 + r + 0.2993262753 + r2 = 0 + 0.2993262753 Reorder the terms: -0.2993262753 + 0.2993262753 + r + r2 = 0 + 0.2993262753 Combine like terms: -0.2993262753 + 0.2993262753 = 0.0000000000 0.0000000000 + r + r2 = 0 + 0.2993262753 r + r2 = 0 + 0.2993262753 Combine like terms: 0 + 0.2993262753 = 0.2993262753 r + r2 = 0.2993262753 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + r2 = 0.2993262753 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 0.2993262753 + 0.25 Combine like terms: 0.2993262753 + 0.25 = 0.5493262753 1.25 + r2 = 0.5493262753 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 0.5493262753 Calculate the square root of the right side: 0.741165484 Break this problem into two subproblems by setting (r + 0.5) equal to 0.741165484 and -0.741165484.Subproblem 1
r + 0.5 = 0.741165484 Simplifying r + 0.5 = 0.741165484 Reorder the terms: 0.5 + r = 0.741165484 Solving 0.5 + r = 0.741165484 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 0.741165484 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.741165484 + -0.5 r = 0.741165484 + -0.5 Combine like terms: 0.741165484 + -0.5 = 0.241165484 r = 0.241165484 Simplifying r = 0.241165484Subproblem 2
r + 0.5 = -0.741165484 Simplifying r + 0.5 = -0.741165484 Reorder the terms: 0.5 + r = -0.741165484 Solving 0.5 + r = -0.741165484 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -0.741165484 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.741165484 + -0.5 r = -0.741165484 + -0.5 Combine like terms: -0.741165484 + -0.5 = -1.241165484 r = -1.241165484 Simplifying r = -1.241165484Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.241165484, -1.241165484}
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