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Simplifying 4r(1 + 32r) = 1 (1 * 4r + 32r * 4r) = 1 (4r + 128r2) = 1 Solving 4r + 128r2 = 1 Solving for variable 'r'. Reorder the terms: -1 + 4r + 128r2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 4r + 128r2 = 0 Begin completing the square. Divide all terms by 128 the coefficient of the squared term: Divide each side by '128'. -0.0078125 + 0.03125r + r2 = 0 Move the constant term to the right: Add '0.0078125' to each side of the equation. -0.0078125 + 0.03125r + 0.0078125 + r2 = 0 + 0.0078125 Reorder the terms: -0.0078125 + 0.0078125 + 0.03125r + r2 = 0 + 0.0078125 Combine like terms: -0.0078125 + 0.0078125 = 0.0000000 0.0000000 + 0.03125r + r2 = 0 + 0.0078125 0.03125r + r2 = 0 + 0.0078125 Combine like terms: 0 + 0.0078125 = 0.0078125 0.03125r + r2 = 0.0078125 The r term is 0.03125r. Take half its coefficient (0.015625). Square it (0.000244140625) and add it to both sides. Add '0.000244140625' to each side of the equation. 0.03125r + 0.000244140625 + r2 = 0.0078125 + 0.000244140625 Reorder the terms: 0.000244140625 + 0.03125r + r2 = 0.0078125 + 0.000244140625 Combine like terms: 0.0078125 + 0.000244140625 = 0.008056640625 0.000244140625 + 0.03125r + r2 = 0.008056640625 Factor a perfect square on the left side: (r + 0.015625)(r + 0.015625) = 0.008056640625 Calculate the square root of the right side: 0.089758791 Break this problem into two subproblems by setting (r + 0.015625) equal to 0.089758791 and -0.089758791.Subproblem 1
r + 0.015625 = 0.089758791 Simplifying r + 0.015625 = 0.089758791 Reorder the terms: 0.015625 + r = 0.089758791 Solving 0.015625 + r = 0.089758791 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.015625' to each side of the equation. 0.015625 + -0.015625 + r = 0.089758791 + -0.015625 Combine like terms: 0.015625 + -0.015625 = 0.000000 0.000000 + r = 0.089758791 + -0.015625 r = 0.089758791 + -0.015625 Combine like terms: 0.089758791 + -0.015625 = 0.074133791 r = 0.074133791 Simplifying r = 0.074133791Subproblem 2
r + 0.015625 = -0.089758791 Simplifying r + 0.015625 = -0.089758791 Reorder the terms: 0.015625 + r = -0.089758791 Solving 0.015625 + r = -0.089758791 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.015625' to each side of the equation. 0.015625 + -0.015625 + r = -0.089758791 + -0.015625 Combine like terms: 0.015625 + -0.015625 = 0.000000 0.000000 + r = -0.089758791 + -0.015625 r = -0.089758791 + -0.015625 Combine like terms: -0.089758791 + -0.015625 = -0.105383791 r = -0.105383791 Simplifying r = -0.105383791Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.074133791, -0.105383791}
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