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Simplifying 453.662 = 2 * 3.14r2 + 2 * 3.14r * 13 Multiply 2 * 3.14 453.662 = 6.28r2 + 2 * 3.14r * 13 Reorder the terms for easier multiplication: 453.662 = 6.28r2 + 2 * 3.14 * 13r Multiply 2 * 3.14 453.662 = 6.28r2 + 6.28 * 13r Multiply 6.28 * 13 453.662 = 6.28r2 + 81.64r Reorder the terms: 453.662 = 81.64r + 6.28r2 Solving 453.662 = 81.64r + 6.28r2 Solving for variable 'r'. Reorder the terms: 453.662 + -81.64r + -6.28r2 = 81.64r + -81.64r + 6.28r2 + -6.28r2 Combine like terms: 81.64r + -81.64r = 0.00 453.662 + -81.64r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 453.662 + -81.64r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 453.662 + -81.64r + -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'. -72.23917197 + 13r + r2 = 0 Move the constant term to the right: Add '72.23917197' to each side of the equation. -72.23917197 + 13r + 72.23917197 + r2 = 0 + 72.23917197 Reorder the terms: -72.23917197 + 72.23917197 + 13r + r2 = 0 + 72.23917197 Combine like terms: -72.23917197 + 72.23917197 = 0.00000000 0.00000000 + 13r + r2 = 0 + 72.23917197 13r + r2 = 0 + 72.23917197 Combine like terms: 0 + 72.23917197 = 72.23917197 13r + r2 = 72.23917197 The r term is 13r. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13r + 42.25 + r2 = 72.23917197 + 42.25 Reorder the terms: 42.25 + 13r + r2 = 72.23917197 + 42.25 Combine like terms: 72.23917197 + 42.25 = 114.48917197 42.25 + 13r + r2 = 114.48917197 Factor a perfect square on the left side: (r + 6.5)(r + 6.5) = 114.48917197 Calculate the square root of the right side: 10.699961307 Break this problem into two subproblems by setting (r + 6.5) equal to 10.699961307 and -10.699961307.Subproblem 1
r + 6.5 = 10.699961307 Simplifying r + 6.5 = 10.699961307 Reorder the terms: 6.5 + r = 10.699961307 Solving 6.5 + r = 10.699961307 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + r = 10.699961307 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + r = 10.699961307 + -6.5 r = 10.699961307 + -6.5 Combine like terms: 10.699961307 + -6.5 = 4.199961307 r = 4.199961307 Simplifying r = 4.199961307Subproblem 2
r + 6.5 = -10.699961307 Simplifying r + 6.5 = -10.699961307 Reorder the terms: 6.5 + r = -10.699961307 Solving 6.5 + r = -10.699961307 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + r = -10.699961307 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + r = -10.699961307 + -6.5 r = -10.699961307 + -6.5 Combine like terms: -10.699961307 + -6.5 = -17.199961307 r = -17.199961307 Simplifying r = -17.199961307Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.199961307, -17.199961307}
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