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Simplifying 6d2 + 33d + 1 = 0 Reorder the terms: 1 + 33d + 6d2 = 0 Solving 1 + 33d + 6d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.1666666667 + 5.5d + d2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + 5.5d + -0.1666666667 + d2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + 5.5d + d2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + 5.5d + d2 = 0 + -0.1666666667 5.5d + d2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 5.5d + d2 = -0.1666666667 The d term is 5.5d. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5d + 7.5625 + d2 = -0.1666666667 + 7.5625 Reorder the terms: 7.5625 + 5.5d + d2 = -0.1666666667 + 7.5625 Combine like terms: -0.1666666667 + 7.5625 = 7.3958333333 7.5625 + 5.5d + d2 = 7.3958333333 Factor a perfect square on the left side: (d + 2.75)(d + 2.75) = 7.3958333333 Calculate the square root of the right side: 2.719528145 Break this problem into two subproblems by setting (d + 2.75) equal to 2.719528145 and -2.719528145.Subproblem 1
d + 2.75 = 2.719528145 Simplifying d + 2.75 = 2.719528145 Reorder the terms: 2.75 + d = 2.719528145 Solving 2.75 + d = 2.719528145 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + d = 2.719528145 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + d = 2.719528145 + -2.75 d = 2.719528145 + -2.75 Combine like terms: 2.719528145 + -2.75 = -0.030471855 d = -0.030471855 Simplifying d = -0.030471855Subproblem 2
d + 2.75 = -2.719528145 Simplifying d + 2.75 = -2.719528145 Reorder the terms: 2.75 + d = -2.719528145 Solving 2.75 + d = -2.719528145 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + d = -2.719528145 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + d = -2.719528145 + -2.75 d = -2.719528145 + -2.75 Combine like terms: -2.719528145 + -2.75 = -5.469528145 d = -5.469528145 Simplifying d = -5.469528145Solution
The solution to the problem is based on the solutions from the subproblems. d = {-0.030471855, -5.469528145}
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