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Simplifying 0.08r + -0.04 = -0.32r2 Reorder the terms: -0.04 + 0.08r = -0.32r2 Solving -0.04 + 0.08r = -0.32r2 Solving for variable 'r'. Combine like terms: -0.32r2 + 0.32r2 = 0.00 -0.04 + 0.08r + 0.32r2 = 0.00 Begin completing the square. Divide all terms by 0.32 the coefficient of the squared term: Divide each side by '0.32'. -0.125 + 0.25r + r2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 0.25r + 0.125 + r2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 0.25r + r2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 0.25r + r2 = 0 + 0.125 0.25r + r2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 0.25r + r2 = 0.125 The r term is 0.25r. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25r + 0.015625 + r2 = 0.125 + 0.015625 Reorder the terms: 0.015625 + 0.25r + r2 = 0.125 + 0.015625 Combine like terms: 0.125 + 0.015625 = 0.140625 0.015625 + 0.25r + r2 = 0.140625 Factor a perfect square on the left side: (r + 0.125)(r + 0.125) = 0.140625 Calculate the square root of the right side: 0.375 Break this problem into two subproblems by setting (r + 0.125) equal to 0.375 and -0.375.Subproblem 1
r + 0.125 = 0.375 Simplifying r + 0.125 = 0.375 Reorder the terms: 0.125 + r = 0.375 Solving 0.125 + r = 0.375 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + r = 0.375 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + r = 0.375 + -0.125 r = 0.375 + -0.125 Combine like terms: 0.375 + -0.125 = 0.25 r = 0.25 Simplifying r = 0.25Subproblem 2
r + 0.125 = -0.375 Simplifying r + 0.125 = -0.375 Reorder the terms: 0.125 + r = -0.375 Solving 0.125 + r = -0.375 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + r = -0.375 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + r = -0.375 + -0.125 r = -0.375 + -0.125 Combine like terms: -0.375 + -0.125 = -0.5 r = -0.5 Simplifying r = -0.5Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.25, -0.5}
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