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Simplifying 0 = 6d2 + -3d + -5 Reorder the terms: 0 = -5 + -3d + 6d2 Solving 0 = -5 + -3d + 6d2 Solving for variable 'd'. Combine like terms: 0 + 5 = 5 5 + 3d + -6d2 = -5 + -3d + 6d2 + 5 + 3d + -6d2 Reorder the terms: 5 + 3d + -6d2 = -5 + 5 + -3d + 3d + 6d2 + -6d2 Combine like terms: -5 + 5 = 0 5 + 3d + -6d2 = 0 + -3d + 3d + 6d2 + -6d2 5 + 3d + -6d2 = -3d + 3d + 6d2 + -6d2 Combine like terms: -3d + 3d = 0 5 + 3d + -6d2 = 0 + 6d2 + -6d2 5 + 3d + -6d2 = 6d2 + -6d2 Combine like terms: 6d2 + -6d2 = 0 5 + 3d + -6d2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -0.8333333333 + -0.5d + d2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + -0.5d + 0.8333333333 + d2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + -0.5d + d2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + -0.5d + d2 = 0 + 0.8333333333 -0.5d + d2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 -0.5d + d2 = 0.8333333333 The d term is -0.5d. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5d + 0.0625 + d2 = 0.8333333333 + 0.0625 Reorder the terms: 0.0625 + -0.5d + d2 = 0.8333333333 + 0.0625 Combine like terms: 0.8333333333 + 0.0625 = 0.8958333333 0.0625 + -0.5d + d2 = 0.8958333333 Factor a perfect square on the left side: (d + -0.25)(d + -0.25) = 0.8958333333 Calculate the square root of the right side: 0.946484724 Break this problem into two subproblems by setting (d + -0.25) equal to 0.946484724 and -0.946484724.Subproblem 1
d + -0.25 = 0.946484724 Simplifying d + -0.25 = 0.946484724 Reorder the terms: -0.25 + d = 0.946484724 Solving -0.25 + d = 0.946484724 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + d = 0.946484724 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + d = 0.946484724 + 0.25 d = 0.946484724 + 0.25 Combine like terms: 0.946484724 + 0.25 = 1.196484724 d = 1.196484724 Simplifying d = 1.196484724Subproblem 2
d + -0.25 = -0.946484724 Simplifying d + -0.25 = -0.946484724 Reorder the terms: -0.25 + d = -0.946484724 Solving -0.25 + d = -0.946484724 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + d = -0.946484724 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + d = -0.946484724 + 0.25 d = -0.946484724 + 0.25 Combine like terms: -0.946484724 + 0.25 = -0.696484724 d = -0.696484724 Simplifying d = -0.696484724Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.196484724, -0.696484724}
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