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Simplifying 6d2 + -21d = 10d + 35 Reorder the terms: -21d + 6d2 = 10d + 35 Reorder the terms: -21d + 6d2 = 35 + 10d Solving -21d + 6d2 = 35 + 10d Solving for variable 'd'. Reorder the terms: -35 + -21d + -10d + 6d2 = 35 + 10d + -35 + -10d Combine like terms: -21d + -10d = -31d -35 + -31d + 6d2 = 35 + 10d + -35 + -10d Reorder the terms: -35 + -31d + 6d2 = 35 + -35 + 10d + -10d Combine like terms: 35 + -35 = 0 -35 + -31d + 6d2 = 0 + 10d + -10d -35 + -31d + 6d2 = 10d + -10d Combine like terms: 10d + -10d = 0 -35 + -31d + 6d2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.833333333 + -5.166666667d + d2 = 0 Move the constant term to the right: Add '5.833333333' to each side of the equation. -5.833333333 + -5.166666667d + 5.833333333 + d2 = 0 + 5.833333333 Reorder the terms: -5.833333333 + 5.833333333 + -5.166666667d + d2 = 0 + 5.833333333 Combine like terms: -5.833333333 + 5.833333333 = 0.000000000 0.000000000 + -5.166666667d + d2 = 0 + 5.833333333 -5.166666667d + d2 = 0 + 5.833333333 Combine like terms: 0 + 5.833333333 = 5.833333333 -5.166666667d + d2 = 5.833333333 The d term is -5.166666667d. Take half its coefficient (-2.583333334). Square it (6.673611115) and add it to both sides. Add '6.673611115' to each side of the equation. -5.166666667d + 6.673611115 + d2 = 5.833333333 + 6.673611115 Reorder the terms: 6.673611115 + -5.166666667d + d2 = 5.833333333 + 6.673611115 Combine like terms: 5.833333333 + 6.673611115 = 12.506944448 6.673611115 + -5.166666667d + d2 = 12.506944448 Factor a perfect square on the left side: (d + -2.583333334)(d + -2.583333334) = 12.506944448 Calculate the square root of the right side: 3.536515863 Break this problem into two subproblems by setting (d + -2.583333334) equal to 3.536515863 and -3.536515863.Subproblem 1
d + -2.583333334 = 3.536515863 Simplifying d + -2.583333334 = 3.536515863 Reorder the terms: -2.583333334 + d = 3.536515863 Solving -2.583333334 + d = 3.536515863 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '2.583333334' to each side of the equation. -2.583333334 + 2.583333334 + d = 3.536515863 + 2.583333334 Combine like terms: -2.583333334 + 2.583333334 = 0.000000000 0.000000000 + d = 3.536515863 + 2.583333334 d = 3.536515863 + 2.583333334 Combine like terms: 3.536515863 + 2.583333334 = 6.119849197 d = 6.119849197 Simplifying d = 6.119849197Subproblem 2
d + -2.583333334 = -3.536515863 Simplifying d + -2.583333334 = -3.536515863 Reorder the terms: -2.583333334 + d = -3.536515863 Solving -2.583333334 + d = -3.536515863 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '2.583333334' to each side of the equation. -2.583333334 + 2.583333334 + d = -3.536515863 + 2.583333334 Combine like terms: -2.583333334 + 2.583333334 = 0.000000000 0.000000000 + d = -3.536515863 + 2.583333334 d = -3.536515863 + 2.583333334 Combine like terms: -3.536515863 + 2.583333334 = -0.953182529 d = -0.953182529 Simplifying d = -0.953182529Solution
The solution to the problem is based on the solutions from the subproblems. d = {6.119849197, -0.953182529}
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