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Simplifying 6.28r2 + -98.5896 = 5092r + 8038.8104 Reorder the terms: -98.5896 + 6.28r2 = 5092r + 8038.8104 Reorder the terms: -98.5896 + 6.28r2 = 8038.8104 + 5092r Solving -98.5896 + 6.28r2 = 8038.8104 + 5092r Solving for variable 'r'. Reorder the terms: -98.5896 + -8038.8104 + -5092r + 6.28r2 = 8038.8104 + 5092r + -8038.8104 + -5092r Combine like terms: -98.5896 + -8038.8104 = -8137.4 -8137.4 + -5092r + 6.28r2 = 8038.8104 + 5092r + -8038.8104 + -5092r Reorder the terms: -8137.4 + -5092r + 6.28r2 = 8038.8104 + -8038.8104 + 5092r + -5092r Combine like terms: 8038.8104 + -8038.8104 = 0.0000 -8137.4 + -5092r + 6.28r2 = 0.0000 + 5092r + -5092r -8137.4 + -5092r + 6.28r2 = 5092r + -5092r Combine like terms: 5092r + -5092r = 0 -8137.4 + -5092r + 6.28r2 = 0 Begin completing the square. Divide all terms by 6.28 the coefficient of the squared term: Divide each side by '6.28'. -1295.764331 + -810.8280255r + r2 = 0 Move the constant term to the right: Add '1295.764331' to each side of the equation. -1295.764331 + -810.8280255r + 1295.764331 + r2 = 0 + 1295.764331 Reorder the terms: -1295.764331 + 1295.764331 + -810.8280255r + r2 = 0 + 1295.764331 Combine like terms: -1295.764331 + 1295.764331 = 0.000000 0.000000 + -810.8280255r + r2 = 0 + 1295.764331 -810.8280255r + r2 = 0 + 1295.764331 Combine like terms: 0 + 1295.764331 = 1295.764331 -810.8280255r + r2 = 1295.764331 The r term is -810.8280255r. Take half its coefficient (-405.4140128). Square it (164360.5218) and add it to both sides. Add '164360.5218' to each side of the equation. -810.8280255r + 164360.5218 + r2 = 1295.764331 + 164360.5218 Reorder the terms: 164360.5218 + -810.8280255r + r2 = 1295.764331 + 164360.5218 Combine like terms: 1295.764331 + 164360.5218 = 165656.286131 164360.5218 + -810.8280255r + r2 = 165656.286131 Factor a perfect square on the left side: (r + -405.4140128)(r + -405.4140128) = 165656.286131 Calculate the square root of the right side: 407.008950922 Break this problem into two subproblems by setting (r + -405.4140128) equal to 407.008950922 and -407.008950922.Subproblem 1
r + -405.4140128 = 407.008950922 Simplifying r + -405.4140128 = 407.008950922 Reorder the terms: -405.4140128 + r = 407.008950922 Solving -405.4140128 + r = 407.008950922 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '405.4140128' to each side of the equation. -405.4140128 + 405.4140128 + r = 407.008950922 + 405.4140128 Combine like terms: -405.4140128 + 405.4140128 = 0.0000000 0.0000000 + r = 407.008950922 + 405.4140128 r = 407.008950922 + 405.4140128 Combine like terms: 407.008950922 + 405.4140128 = 812.422963722 r = 812.422963722 Simplifying r = 812.422963722Subproblem 2
r + -405.4140128 = -407.008950922 Simplifying r + -405.4140128 = -407.008950922 Reorder the terms: -405.4140128 + r = -407.008950922 Solving -405.4140128 + r = -407.008950922 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '405.4140128' to each side of the equation. -405.4140128 + 405.4140128 + r = -407.008950922 + 405.4140128 Combine like terms: -405.4140128 + 405.4140128 = 0.0000000 0.0000000 + r = -407.008950922 + 405.4140128 r = -407.008950922 + 405.4140128 Combine like terms: -407.008950922 + 405.4140128 = -1.594938122 r = -1.594938122 Simplifying r = -1.594938122Solution
The solution to the problem is based on the solutions from the subproblems. r = {812.422963722, -1.594938122}
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