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Simplifying (5d + -2) + (6d(6d) + 4d) + -1(2d + -1) = 0 Reorder the terms: (-2 + 5d) + (6d(6d) + 4d) + -1(2d + -1) = 0 Remove parenthesis around (-2 + 5d) -2 + 5d + (6d(6d) + 4d) + -1(2d + -1) = 0 Remove parenthesis around (6d) -2 + 5d + (6d * 6d + 4d) + -1(2d + -1) = 0 Reorder the terms for easier multiplication: -2 + 5d + (6 * 6d * d + 4d) + -1(2d + -1) = 0 Multiply 6 * 6 -2 + 5d + (36d * d + 4d) + -1(2d + -1) = 0 Multiply d * d -2 + 5d + (36d2 + 4d) + -1(2d + -1) = 0 Reorder the terms: -2 + 5d + (4d + 36d2) + -1(2d + -1) = 0 Remove parenthesis around (4d + 36d2) -2 + 5d + 4d + 36d2 + -1(2d + -1) = 0 Reorder the terms: -2 + 5d + 4d + 36d2 + -1(-1 + 2d) = 0 -2 + 5d + 4d + 36d2 + (-1 * -1 + 2d * -1) = 0 -2 + 5d + 4d + 36d2 + (1 + -2d) = 0 Reorder the terms: -2 + 1 + 5d + 4d + -2d + 36d2 = 0 Combine like terms: -2 + 1 = -1 -1 + 5d + 4d + -2d + 36d2 = 0 Combine like terms: 5d + 4d = 9d -1 + 9d + -2d + 36d2 = 0 Combine like terms: 9d + -2d = 7d -1 + 7d + 36d2 = 0 Solving -1 + 7d + 36d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. -0.02777777778 + 0.1944444444d + d2 = 0 Move the constant term to the right: Add '0.02777777778' to each side of the equation. -0.02777777778 + 0.1944444444d + 0.02777777778 + d2 = 0 + 0.02777777778 Reorder the terms: -0.02777777778 + 0.02777777778 + 0.1944444444d + d2 = 0 + 0.02777777778 Combine like terms: -0.02777777778 + 0.02777777778 = 0.00000000000 0.00000000000 + 0.1944444444d + d2 = 0 + 0.02777777778 0.1944444444d + d2 = 0 + 0.02777777778 Combine like terms: 0 + 0.02777777778 = 0.02777777778 0.1944444444d + d2 = 0.02777777778 The d term is 0.1944444444d. Take half its coefficient (0.0972222222). Square it (0.009452160490) and add it to both sides. Add '0.009452160490' to each side of the equation. 0.1944444444d + 0.009452160490 + d2 = 0.02777777778 + 0.009452160490 Reorder the terms: 0.009452160490 + 0.1944444444d + d2 = 0.02777777778 + 0.009452160490 Combine like terms: 0.02777777778 + 0.009452160490 = 0.03722993827 0.009452160490 + 0.1944444444d + d2 = 0.03722993827 Factor a perfect square on the left side: (d + 0.0972222222)(d + 0.0972222222) = 0.03722993827 Calculate the square root of the right side: 0.192950611 Break this problem into two subproblems by setting (d + 0.0972222222) equal to 0.192950611 and -0.192950611.Subproblem 1
d + 0.0972222222 = 0.192950611 Simplifying d + 0.0972222222 = 0.192950611 Reorder the terms: 0.0972222222 + d = 0.192950611 Solving 0.0972222222 + d = 0.192950611 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.0972222222' to each side of the equation. 0.0972222222 + -0.0972222222 + d = 0.192950611 + -0.0972222222 Combine like terms: 0.0972222222 + -0.0972222222 = 0.0000000000 0.0000000000 + d = 0.192950611 + -0.0972222222 d = 0.192950611 + -0.0972222222 Combine like terms: 0.192950611 + -0.0972222222 = 0.0957283888 d = 0.0957283888 Simplifying d = 0.0957283888Subproblem 2
d + 0.0972222222 = -0.192950611 Simplifying d + 0.0972222222 = -0.192950611 Reorder the terms: 0.0972222222 + d = -0.192950611 Solving 0.0972222222 + d = -0.192950611 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.0972222222' to each side of the equation. 0.0972222222 + -0.0972222222 + d = -0.192950611 + -0.0972222222 Combine like terms: 0.0972222222 + -0.0972222222 = 0.0000000000 0.0000000000 + d = -0.192950611 + -0.0972222222 d = -0.192950611 + -0.0972222222 Combine like terms: -0.192950611 + -0.0972222222 = -0.2901728332 d = -0.2901728332 Simplifying d = -0.2901728332Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.0957283888, -0.2901728332}
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