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Simplifying 512.3852 = 2 * 3.14r2 + 2 * 3.14r * 16.2 Multiply 2 * 3.14 512.3852 = 6.28r2 + 2 * 3.14r * 16.2 Reorder the terms for easier multiplication: 512.3852 = 6.28r2 + 2 * 3.14 * 16.2r Multiply 2 * 3.14 512.3852 = 6.28r2 + 6.28 * 16.2r Multiply 6.28 * 16.2 512.3852 = 6.28r2 + 101.736r Reorder the terms: 512.3852 = 101.736r + 6.28r2 Solving 512.3852 = 101.736r + 6.28r2 Solving for variable 'r'. Reorder the terms: 512.3852 + -101.736r + -6.28r2 = 101.736r + -101.736r + 6.28r2 + -6.28r2 Combine like terms: 101.736r + -101.736r = 0.000 512.3852 + -101.736r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 512.3852 + -101.736r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 512.3852 + -101.736r + -6.28r2 = 0.00 Begin completing the square. Divide all terms by -6.28 the coefficient of the squared term: Divide each side by '-6.28'. -81.59 + 16.2r + r2 = 0 Move the constant term to the right: Add '81.59' to each side of the equation. -81.59 + 16.2r + 81.59 + r2 = 0 + 81.59 Reorder the terms: -81.59 + 81.59 + 16.2r + r2 = 0 + 81.59 Combine like terms: -81.59 + 81.59 = 0.00 0.00 + 16.2r + r2 = 0 + 81.59 16.2r + r2 = 0 + 81.59 Combine like terms: 0 + 81.59 = 81.59 16.2r + r2 = 81.59 The r term is 16.2r. Take half its coefficient (8.1). Square it (65.61) and add it to both sides. Add '65.61' to each side of the equation. 16.2r + 65.61 + r2 = 81.59 + 65.61 Reorder the terms: 65.61 + 16.2r + r2 = 81.59 + 65.61 Combine like terms: 81.59 + 65.61 = 147.2 65.61 + 16.2r + r2 = 147.2 Factor a perfect square on the left side: (r + 8.1)(r + 8.1) = 147.2 Calculate the square root of the right side: 12.13260071 Break this problem into two subproblems by setting (r + 8.1) equal to 12.13260071 and -12.13260071.Subproblem 1
r + 8.1 = 12.13260071 Simplifying r + 8.1 = 12.13260071 Reorder the terms: 8.1 + r = 12.13260071 Solving 8.1 + r = 12.13260071 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.1' to each side of the equation. 8.1 + -8.1 + r = 12.13260071 + -8.1 Combine like terms: 8.1 + -8.1 = 0.0 0.0 + r = 12.13260071 + -8.1 r = 12.13260071 + -8.1 Combine like terms: 12.13260071 + -8.1 = 4.03260071 r = 4.03260071 Simplifying r = 4.03260071Subproblem 2
r + 8.1 = -12.13260071 Simplifying r + 8.1 = -12.13260071 Reorder the terms: 8.1 + r = -12.13260071 Solving 8.1 + r = -12.13260071 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.1' to each side of the equation. 8.1 + -8.1 + r = -12.13260071 + -8.1 Combine like terms: 8.1 + -8.1 = 0.0 0.0 + r = -12.13260071 + -8.1 r = -12.13260071 + -8.1 Combine like terms: -12.13260071 + -8.1 = -20.23260071 r = -20.23260071 Simplifying r = -20.23260071Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.03260071, -20.23260071}
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