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Simplifying 0 = -200r2 + 92000r + -8400000 Reorder the terms: 0 = -8400000 + 92000r + -200r2 Solving 0 = -8400000 + 92000r + -200r2 Solving for variable 'r'. Combine like terms: 0 + 8400000 = 8400000 8400000 + -92000r + 200r2 = -8400000 + 92000r + -200r2 + 8400000 + -92000r + 200r2 Reorder the terms: 8400000 + -92000r + 200r2 = -8400000 + 8400000 + 92000r + -92000r + -200r2 + 200r2 Combine like terms: -8400000 + 8400000 = 0 8400000 + -92000r + 200r2 = 0 + 92000r + -92000r + -200r2 + 200r2 8400000 + -92000r + 200r2 = 92000r + -92000r + -200r2 + 200r2 Combine like terms: 92000r + -92000r = 0 8400000 + -92000r + 200r2 = 0 + -200r2 + 200r2 8400000 + -92000r + 200r2 = -200r2 + 200r2 Combine like terms: -200r2 + 200r2 = 0 8400000 + -92000r + 200r2 = 0 Factor out the Greatest Common Factor (GCF), '200'. 200(42000 + -460r + r2) = 0 Ignore the factor 200.Subproblem 1
Set the factor '(42000 + -460r + r2)' equal to zero and attempt to solve: Simplifying 42000 + -460r + r2 = 0 Solving 42000 + -460r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '-42000' to each side of the equation. 42000 + -460r + -42000 + r2 = 0 + -42000 Reorder the terms: 42000 + -42000 + -460r + r2 = 0 + -42000 Combine like terms: 42000 + -42000 = 0 0 + -460r + r2 = 0 + -42000 -460r + r2 = 0 + -42000 Combine like terms: 0 + -42000 = -42000 -460r + r2 = -42000 The r term is -460r. Take half its coefficient (-230). Square it (52900) and add it to both sides. Add '52900' to each side of the equation. -460r + 52900 + r2 = -42000 + 52900 Reorder the terms: 52900 + -460r + r2 = -42000 + 52900 Combine like terms: -42000 + 52900 = 10900 52900 + -460r + r2 = 10900 Factor a perfect square on the left side: (r + -230)(r + -230) = 10900 Calculate the square root of the right side: 104.403065089 Break this problem into two subproblems by setting (r + -230) equal to 104.403065089 and -104.403065089.Subproblem 1
r + -230 = 104.403065089 Simplifying r + -230 = 104.403065089 Reorder the terms: -230 + r = 104.403065089 Solving -230 + r = 104.403065089 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '230' to each side of the equation. -230 + 230 + r = 104.403065089 + 230 Combine like terms: -230 + 230 = 0 0 + r = 104.403065089 + 230 r = 104.403065089 + 230 Combine like terms: 104.403065089 + 230 = 334.403065089 r = 334.403065089 Simplifying r = 334.403065089Subproblem 2
r + -230 = -104.403065089 Simplifying r + -230 = -104.403065089 Reorder the terms: -230 + r = -104.403065089 Solving -230 + r = -104.403065089 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '230' to each side of the equation. -230 + 230 + r = -104.403065089 + 230 Combine like terms: -230 + 230 = 0 0 + r = -104.403065089 + 230 r = -104.403065089 + 230 Combine like terms: -104.403065089 + 230 = 125.596934911 r = 125.596934911 Simplifying r = 125.596934911Solution
The solution to the problem is based on the solutions from the subproblems. r = {334.403065089, 125.596934911}Solution
r = {334.403065089, 125.596934911}
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