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Simplifying 40000 = 5301.4D2 + 34871.7D Reorder the terms: 40000 = 34871.7D + 5301.4D2 Solving 40000 = 34871.7D + 5301.4D2 Solving for variable 'D'. Reorder the terms: 40000 + -34871.7D + -5301.4D2 = 34871.7D + -34871.7D + 5301.4D2 + -5301.4D2 Combine like terms: 34871.7D + -34871.7D = 0.0 40000 + -34871.7D + -5301.4D2 = 0.0 + 5301.4D2 + -5301.4D2 40000 + -34871.7D + -5301.4D2 = 5301.4D2 + -5301.4D2 Combine like terms: 5301.4D2 + -5301.4D2 = 0.0 40000 + -34871.7D + -5301.4D2 = 0.0 Begin completing the square. Divide all terms by -5301.4 the coefficient of the squared term: Divide each side by '-5301.4'. -7.545176746 + 6.577828498D + D2 = 0 Move the constant term to the right: Add '7.545176746' to each side of the equation. -7.545176746 + 6.577828498D + 7.545176746 + D2 = 0 + 7.545176746 Reorder the terms: -7.545176746 + 7.545176746 + 6.577828498D + D2 = 0 + 7.545176746 Combine like terms: -7.545176746 + 7.545176746 = 0.000000000 0.000000000 + 6.577828498D + D2 = 0 + 7.545176746 6.577828498D + D2 = 0 + 7.545176746 Combine like terms: 0 + 7.545176746 = 7.545176746 6.577828498D + D2 = 7.545176746 The D term is 6.577828498D. Take half its coefficient (3.288914249). Square it (10.81695694) and add it to both sides. Add '10.81695694' to each side of the equation. 6.577828498D + 10.81695694 + D2 = 7.545176746 + 10.81695694 Reorder the terms: 10.81695694 + 6.577828498D + D2 = 7.545176746 + 10.81695694 Combine like terms: 7.545176746 + 10.81695694 = 18.362133686 10.81695694 + 6.577828498D + D2 = 18.362133686 Factor a perfect square on the left side: (D + 3.288914249)(D + 3.288914249) = 18.362133686 Calculate the square root of the right side: 4.28510603 Break this problem into two subproblems by setting (D + 3.288914249) equal to 4.28510603 and -4.28510603.Subproblem 1
D + 3.288914249 = 4.28510603 Simplifying D + 3.288914249 = 4.28510603 Reorder the terms: 3.288914249 + D = 4.28510603 Solving 3.288914249 + D = 4.28510603 Solving for variable 'D'. Move all terms containing D to the left, all other terms to the right. Add '-3.288914249' to each side of the equation. 3.288914249 + -3.288914249 + D = 4.28510603 + -3.288914249 Combine like terms: 3.288914249 + -3.288914249 = 0.000000000 0.000000000 + D = 4.28510603 + -3.288914249 D = 4.28510603 + -3.288914249 Combine like terms: 4.28510603 + -3.288914249 = 0.996191781 D = 0.996191781 Simplifying D = 0.996191781Subproblem 2
D + 3.288914249 = -4.28510603 Simplifying D + 3.288914249 = -4.28510603 Reorder the terms: 3.288914249 + D = -4.28510603 Solving 3.288914249 + D = -4.28510603 Solving for variable 'D'. Move all terms containing D to the left, all other terms to the right. Add '-3.288914249' to each side of the equation. 3.288914249 + -3.288914249 + D = -4.28510603 + -3.288914249 Combine like terms: 3.288914249 + -3.288914249 = 0.000000000 0.000000000 + D = -4.28510603 + -3.288914249 D = -4.28510603 + -3.288914249 Combine like terms: -4.28510603 + -3.288914249 = -7.574020279 D = -7.574020279 Simplifying D = -7.574020279Solution
The solution to the problem is based on the solutions from the subproblems. D = {0.996191781, -7.574020279}
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