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Simplifying 2010 = 6.28r2 + 6.28r * 36.5 Reorder the terms for easier multiplication: 2010 = 6.28r2 + 6.28 * 36.5r Multiply 6.28 * 36.5 2010 = 6.28r2 + 229.22r Reorder the terms: 2010 = 229.22r + 6.28r2 Solving 2010 = 229.22r + 6.28r2 Solving for variable 'r'. Reorder the terms: 2010 + -229.22r + -6.28r2 = 229.22r + -229.22r + 6.28r2 + -6.28r2 Combine like terms: 229.22r + -229.22r = 0.00 2010 + -229.22r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 2010 + -229.22r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 2010 + -229.22r + -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'. -320.0636943 + 36.5r + r2 = 0 Move the constant term to the right: Add '320.0636943' to each side of the equation. -320.0636943 + 36.5r + 320.0636943 + r2 = 0 + 320.0636943 Reorder the terms: -320.0636943 + 320.0636943 + 36.5r + r2 = 0 + 320.0636943 Combine like terms: -320.0636943 + 320.0636943 = 0.0000000 0.0000000 + 36.5r + r2 = 0 + 320.0636943 36.5r + r2 = 0 + 320.0636943 Combine like terms: 0 + 320.0636943 = 320.0636943 36.5r + r2 = 320.0636943 The r term is 36.5r. Take half its coefficient (18.25). Square it (333.0625) and add it to both sides. Add '333.0625' to each side of the equation. 36.5r + 333.0625 + r2 = 320.0636943 + 333.0625 Reorder the terms: 333.0625 + 36.5r + r2 = 320.0636943 + 333.0625 Combine like terms: 320.0636943 + 333.0625 = 653.1261943 333.0625 + 36.5r + r2 = 653.1261943 Factor a perfect square on the left side: (r + 18.25)(r + 18.25) = 653.1261943 Calculate the square root of the right side: 25.556333741 Break this problem into two subproblems by setting (r + 18.25) equal to 25.556333741 and -25.556333741.Subproblem 1
r + 18.25 = 25.556333741 Simplifying r + 18.25 = 25.556333741 Reorder the terms: 18.25 + r = 25.556333741 Solving 18.25 + r = 25.556333741 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-18.25' to each side of the equation. 18.25 + -18.25 + r = 25.556333741 + -18.25 Combine like terms: 18.25 + -18.25 = 0.00 0.00 + r = 25.556333741 + -18.25 r = 25.556333741 + -18.25 Combine like terms: 25.556333741 + -18.25 = 7.306333741 r = 7.306333741 Simplifying r = 7.306333741Subproblem 2
r + 18.25 = -25.556333741 Simplifying r + 18.25 = -25.556333741 Reorder the terms: 18.25 + r = -25.556333741 Solving 18.25 + r = -25.556333741 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-18.25' to each side of the equation. 18.25 + -18.25 + r = -25.556333741 + -18.25 Combine like terms: 18.25 + -18.25 = 0.00 0.00 + r = -25.556333741 + -18.25 r = -25.556333741 + -18.25 Combine like terms: -25.556333741 + -18.25 = -43.806333741 r = -43.806333741 Simplifying r = -43.806333741Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.306333741, -43.806333741}
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