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Simplifying 2180000 = -200r2 + 92000r + -840000 Reorder the terms: 2180000 = -840000 + 92000r + -200r2 Solving 2180000 = -840000 + 92000r + -200r2 Solving for variable 'r'. Combine like terms: 2180000 + 840000 = 3020000 3020000 + -92000r + 200r2 = -840000 + 92000r + -200r2 + 840000 + -92000r + 200r2 Reorder the terms: 3020000 + -92000r + 200r2 = -840000 + 840000 + 92000r + -92000r + -200r2 + 200r2 Combine like terms: -840000 + 840000 = 0 3020000 + -92000r + 200r2 = 0 + 92000r + -92000r + -200r2 + 200r2 3020000 + -92000r + 200r2 = 92000r + -92000r + -200r2 + 200r2 Combine like terms: 92000r + -92000r = 0 3020000 + -92000r + 200r2 = 0 + -200r2 + 200r2 3020000 + -92000r + 200r2 = -200r2 + 200r2 Combine like terms: -200r2 + 200r2 = 0 3020000 + -92000r + 200r2 = 0 Factor out the Greatest Common Factor (GCF), '200'. 200(15100 + -460r + r2) = 0 Ignore the factor 200.Subproblem 1
Set the factor '(15100 + -460r + r2)' equal to zero and attempt to solve: Simplifying 15100 + -460r + r2 = 0 Solving 15100 + -460r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '-15100' to each side of the equation. 15100 + -460r + -15100 + r2 = 0 + -15100 Reorder the terms: 15100 + -15100 + -460r + r2 = 0 + -15100 Combine like terms: 15100 + -15100 = 0 0 + -460r + r2 = 0 + -15100 -460r + r2 = 0 + -15100 Combine like terms: 0 + -15100 = -15100 -460r + r2 = -15100 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 = -15100 + 52900 Reorder the terms: 52900 + -460r + r2 = -15100 + 52900 Combine like terms: -15100 + 52900 = 37800 52900 + -460r + r2 = 37800 Factor a perfect square on the left side: (r + -230)(r + -230) = 37800 Calculate the square root of the right side: 194.422220952 Break this problem into two subproblems by setting (r + -230) equal to 194.422220952 and -194.422220952.Subproblem 1
r + -230 = 194.422220952 Simplifying r + -230 = 194.422220952 Reorder the terms: -230 + r = 194.422220952 Solving -230 + r = 194.422220952 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 = 194.422220952 + 230 Combine like terms: -230 + 230 = 0 0 + r = 194.422220952 + 230 r = 194.422220952 + 230 Combine like terms: 194.422220952 + 230 = 424.422220952 r = 424.422220952 Simplifying r = 424.422220952Subproblem 2
r + -230 = -194.422220952 Simplifying r + -230 = -194.422220952 Reorder the terms: -230 + r = -194.422220952 Solving -230 + r = -194.422220952 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 = -194.422220952 + 230 Combine like terms: -230 + 230 = 0 0 + r = -194.422220952 + 230 r = -194.422220952 + 230 Combine like terms: -194.422220952 + 230 = 35.577779048 r = 35.577779048 Simplifying r = 35.577779048Solution
The solution to the problem is based on the solutions from the subproblems. r = {424.422220952, 35.577779048}Solution
r = {424.422220952, 35.577779048}
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