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Simplifying 0 = 6.28r2 + 2 + -61273.372r + -3087.308736 Reorder the terms: 0 = 2 + -3087.308736 + -61273.372r + 6.28r2 Combine like terms: 2 + -3087.308736 = -3085.308736 0 = -3085.308736 + -61273.372r + 6.28r2 Solving 0 = -3085.308736 + -61273.372r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 3085.308736 = 3085.308736 3085.308736 + 61273.372r + -6.28r2 = -3085.308736 + -61273.372r + 6.28r2 + 3085.308736 + 61273.372r + -6.28r2 Reorder the terms: 3085.308736 + 61273.372r + -6.28r2 = -3085.308736 + 3085.308736 + -61273.372r + 61273.372r + 6.28r2 + -6.28r2 Combine like terms: -3085.308736 + 3085.308736 = 0.000000 3085.308736 + 61273.372r + -6.28r2 = 0.000000 + -61273.372r + 61273.372r + 6.28r2 + -6.28r2 3085.308736 + 61273.372r + -6.28r2 = -61273.372r + 61273.372r + 6.28r2 + -6.28r2 Combine like terms: -61273.372r + 61273.372r = 0.000 3085.308736 + 61273.372r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 3085.308736 + 61273.372r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 3085.308736 + 61273.372r + -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'. -491.2912 + -9756.906369r + r2 = 0 Move the constant term to the right: Add '491.2912' to each side of the equation. -491.2912 + -9756.906369r + 491.2912 + r2 = 0 + 491.2912 Reorder the terms: -491.2912 + 491.2912 + -9756.906369r + r2 = 0 + 491.2912 Combine like terms: -491.2912 + 491.2912 = 0.0000 0.0000 + -9756.906369r + r2 = 0 + 491.2912 -9756.906369r + r2 = 0 + 491.2912 Combine like terms: 0 + 491.2912 = 491.2912 -9756.906369r + r2 = 491.2912 The r term is -9756.906369r. Take half its coefficient (-4878.453185). Square it (23799305.48) and add it to both sides. Add '23799305.48' to each side of the equation. -9756.906369r + 23799305.48 + r2 = 491.2912 + 23799305.48 Reorder the terms: 23799305.48 + -9756.906369r + r2 = 491.2912 + 23799305.48 Combine like terms: 491.2912 + 23799305.48 = 23799796.7712 23799305.48 + -9756.906369r + r2 = 23799796.7712 Factor a perfect square on the left side: (r + -4878.453185)(r + -4878.453185) = 23799796.7712 Calculate the square root of the right side: 4878.503538094 Break this problem into two subproblems by setting (r + -4878.453185) equal to 4878.503538094 and -4878.503538094.Subproblem 1
r + -4878.453185 = 4878.503538094 Simplifying r + -4878.453185 = 4878.503538094 Reorder the terms: -4878.453185 + r = 4878.503538094 Solving -4878.453185 + r = 4878.503538094 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4878.453185' to each side of the equation. -4878.453185 + 4878.453185 + r = 4878.503538094 + 4878.453185 Combine like terms: -4878.453185 + 4878.453185 = 0.000000 0.000000 + r = 4878.503538094 + 4878.453185 r = 4878.503538094 + 4878.453185 Combine like terms: 4878.503538094 + 4878.453185 = 9756.956723094 r = 9756.956723094 Simplifying r = 9756.956723094Subproblem 2
r + -4878.453185 = -4878.503538094 Simplifying r + -4878.453185 = -4878.503538094 Reorder the terms: -4878.453185 + r = -4878.503538094 Solving -4878.453185 + r = -4878.503538094 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4878.453185' to each side of the equation. -4878.453185 + 4878.453185 + r = -4878.503538094 + 4878.453185 Combine like terms: -4878.453185 + 4878.453185 = 0.000000 0.000000 + r = -4878.503538094 + 4878.453185 r = -4878.503538094 + 4878.453185 Combine like terms: -4878.503538094 + 4878.453185 = -0.050353094 r = -0.050353094 Simplifying r = -0.050353094Solution
The solution to the problem is based on the solutions from the subproblems. r = {9756.956723094, -0.050353094}
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