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Simplifying 2184.05 = 2000(1 + r)(1 + r) Multiply (1 + r) * (1 + r) 2184.05 = 2000(1(1 + r) + r(1 + r)) 2184.05 = 2000((1 * 1 + r * 1) + r(1 + r)) 2184.05 = 2000((1 + 1r) + r(1 + r)) 2184.05 = 2000(1 + 1r + (1 * r + r * r)) 2184.05 = 2000(1 + 1r + (1r + r2)) Combine like terms: 1r + 1r = 2r 2184.05 = 2000(1 + 2r + r2) 2184.05 = (1 * 2000 + 2r * 2000 + r2 * 2000) 2184.05 = (2000 + 4000r + 2000r2) Solving 2184.05 = 2000 + 4000r + 2000r2 Solving for variable 'r'. Combine like terms: 2184.05 + -2000 = 184.05 184.05 + -4000r + -2000r2 = 2000 + 4000r + 2000r2 + -2000 + -4000r + -2000r2 Reorder the terms: 184.05 + -4000r + -2000r2 = 2000 + -2000 + 4000r + -4000r + 2000r2 + -2000r2 Combine like terms: 2000 + -2000 = 0 184.05 + -4000r + -2000r2 = 0 + 4000r + -4000r + 2000r2 + -2000r2 184.05 + -4000r + -2000r2 = 4000r + -4000r + 2000r2 + -2000r2 Combine like terms: 4000r + -4000r = 0 184.05 + -4000r + -2000r2 = 0 + 2000r2 + -2000r2 184.05 + -4000r + -2000r2 = 2000r2 + -2000r2 Combine like terms: 2000r2 + -2000r2 = 0 184.05 + -4000r + -2000r2 = 0 Begin completing the square. Divide all terms by -2000 the coefficient of the squared term: Divide each side by '-2000'. -0.092025 + 2r + r2 = 0 Move the constant term to the right: Add '0.092025' to each side of the equation. -0.092025 + 2r + 0.092025 + r2 = 0 + 0.092025 Reorder the terms: -0.092025 + 0.092025 + 2r + r2 = 0 + 0.092025 Combine like terms: -0.092025 + 0.092025 = 0.000000 0.000000 + 2r + r2 = 0 + 0.092025 2r + r2 = 0 + 0.092025 Combine like terms: 0 + 0.092025 = 0.092025 2r + r2 = 0.092025 The r term is 2r. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2r + 1 + r2 = 0.092025 + 1 Reorder the terms: 1 + 2r + r2 = 0.092025 + 1 Combine like terms: 0.092025 + 1 = 1.092025 1 + 2r + r2 = 1.092025 Factor a perfect square on the left side: (r + 1)(r + 1) = 1.092025 Calculate the square root of the right side: 1.045 Break this problem into two subproblems by setting (r + 1) equal to 1.045 and -1.045.Subproblem 1
r + 1 = 1.045 Simplifying r + 1 = 1.045 Reorder the terms: 1 + r = 1.045 Solving 1 + r = 1.045 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 1.045 + -1 Combine like terms: 1 + -1 = 0 0 + r = 1.045 + -1 r = 1.045 + -1 Combine like terms: 1.045 + -1 = 0.045 r = 0.045 Simplifying r = 0.045Subproblem 2
r + 1 = -1.045 Simplifying r + 1 = -1.045 Reorder the terms: 1 + r = -1.045 Solving 1 + r = -1.045 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -1.045 + -1 Combine like terms: 1 + -1 = 0 0 + r = -1.045 + -1 r = -1.045 + -1 Combine like terms: -1.045 + -1 = -2.045 r = -2.045 Simplifying r = -2.045Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.045, -2.045}
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