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Simplifying 0 = 6.28r2 + 6.28r + -8038.4 Reorder the terms: 0 = -8038.4 + 6.28r + 6.28r2 Solving 0 = -8038.4 + 6.28r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 8038.4 = 8038.4 8038.4 + -6.28r + -6.28r2 = -8038.4 + 6.28r + 6.28r2 + 8038.4 + -6.28r + -6.28r2 Reorder the terms: 8038.4 + -6.28r + -6.28r2 = -8038.4 + 8038.4 + 6.28r + -6.28r + 6.28r2 + -6.28r2 Combine like terms: -8038.4 + 8038.4 = 0.0 8038.4 + -6.28r + -6.28r2 = 0.0 + 6.28r + -6.28r + 6.28r2 + -6.28r2 8038.4 + -6.28r + -6.28r2 = 6.28r + -6.28r + 6.28r2 + -6.28r2 Combine like terms: 6.28r + -6.28r = 0.00 8038.4 + -6.28r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 8038.4 + -6.28r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 8038.4 + -6.28r + -6.28r2 = 0.00 Begin completing the square. Divide all terms by -6.28 the coefficient of the squared term: Divide each side by '-6.28'. -1280 + r + r2 = 0 Move the constant term to the right: Add '1280' to each side of the equation. -1280 + r + 1280 + r2 = 0 + 1280 Reorder the terms: -1280 + 1280 + r + r2 = 0 + 1280 Combine like terms: -1280 + 1280 = 0 0 + r + r2 = 0 + 1280 r + r2 = 0 + 1280 Combine like terms: 0 + 1280 = 1280 r + r2 = 1280 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 = 1280 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 1280 + 0.25 Combine like terms: 1280 + 0.25 = 1280.25 1.25 + r2 = 1280.25 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 1280.25 Calculate the square root of the right side: 35.780581326 Break this problem into two subproblems by setting (r + 0.5) equal to 35.780581326 and -35.780581326.Subproblem 1
r + 0.5 = 35.780581326 Simplifying r + 0.5 = 35.780581326 Reorder the terms: 0.5 + r = 35.780581326 Solving 0.5 + r = 35.780581326 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 = 35.780581326 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 35.780581326 + -0.5 r = 35.780581326 + -0.5 Combine like terms: 35.780581326 + -0.5 = 35.280581326 r = 35.280581326 Simplifying r = 35.280581326Subproblem 2
r + 0.5 = -35.780581326 Simplifying r + 0.5 = -35.780581326 Reorder the terms: 0.5 + r = -35.780581326 Solving 0.5 + r = -35.780581326 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 = -35.780581326 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -35.780581326 + -0.5 r = -35.780581326 + -0.5 Combine like terms: -35.780581326 + -0.5 = -36.280581326 r = -36.280581326 Simplifying r = -36.280581326Solution
The solution to the problem is based on the solutions from the subproblems. r = {35.280581326, -36.280581326}
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