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Simplifying 12.8000 = 0.0009(6.28r2 + 1130.40r) Reorder the terms: 12.8000 = 0.0009(1130.40r + 6.28r2) 12.8000 = (1130.40r * 0.0009 + 6.28r2 * 0.0009) 12.8000 = (1.01736r + 0.005652r2) Solving 12.8000 = 1.01736r + 0.005652r2 Solving for variable 'r'. Reorder the terms: 12.8000 + -1.01736r + -0.005652r2 = 1.01736r + -1.01736r + 0.005652r2 + -0.005652r2 Combine like terms: 1.01736r + -1.01736r = 0.00000 12.8000 + -1.01736r + -0.005652r2 = 0.00000 + 0.005652r2 + -0.005652r2 12.8000 + -1.01736r + -0.005652r2 = 0.005652r2 + -0.005652r2 Combine like terms: 0.005652r2 + -0.005652r2 = 0.000000 12.8000 + -1.01736r + -0.005652r2 = 0.000000 Begin completing the square. Divide all terms by -0.005652 the coefficient of the squared term: Divide each side by '-0.005652'. -2264.685067 + 180r + r2 = 0 Move the constant term to the right: Add '2264.685067' to each side of the equation. -2264.685067 + 180r + 2264.685067 + r2 = 0 + 2264.685067 Reorder the terms: -2264.685067 + 2264.685067 + 180r + r2 = 0 + 2264.685067 Combine like terms: -2264.685067 + 2264.685067 = 0.000000 0.000000 + 180r + r2 = 0 + 2264.685067 180r + r2 = 0 + 2264.685067 Combine like terms: 0 + 2264.685067 = 2264.685067 180r + r2 = 2264.685067 The r term is 180r. Take half its coefficient (90). Square it (8100) and add it to both sides. Add '8100' to each side of the equation. 180r + 8100 + r2 = 2264.685067 + 8100 Reorder the terms: 8100 + 180r + r2 = 2264.685067 + 8100 Combine like terms: 2264.685067 + 8100 = 10364.685067 8100 + 180r + r2 = 10364.685067 Factor a perfect square on the left side: (r + 90)(r + 90) = 10364.685067 Calculate the square root of the right side: 101.807097331 Break this problem into two subproblems by setting (r + 90) equal to 101.807097331 and -101.807097331.Subproblem 1
r + 90 = 101.807097331 Simplifying r + 90 = 101.807097331 Reorder the terms: 90 + r = 101.807097331 Solving 90 + r = 101.807097331 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-90' to each side of the equation. 90 + -90 + r = 101.807097331 + -90 Combine like terms: 90 + -90 = 0 0 + r = 101.807097331 + -90 r = 101.807097331 + -90 Combine like terms: 101.807097331 + -90 = 11.807097331 r = 11.807097331 Simplifying r = 11.807097331Subproblem 2
r + 90 = -101.807097331 Simplifying r + 90 = -101.807097331 Reorder the terms: 90 + r = -101.807097331 Solving 90 + r = -101.807097331 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-90' to each side of the equation. 90 + -90 + r = -101.807097331 + -90 Combine like terms: 90 + -90 = 0 0 + r = -101.807097331 + -90 r = -101.807097331 + -90 Combine like terms: -101.807097331 + -90 = -191.807097331 r = -191.807097331 Simplifying r = -191.807097331Solution
The solution to the problem is based on the solutions from the subproblems. r = {11.807097331, -191.807097331}
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