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Simplifying -4d3 + 28d2 + -4d = 0 Reorder the terms: -4d + 28d2 + -4d3 = 0 Solving -4d + 28d2 + -4d3 = 0 Solving for variable 'd'. Factor out the Greatest Common Factor (GCF), '4d'. 4d(-1 + 7d + -1d2) = 0 Ignore the factor 4.Subproblem 1
Set the factor 'd' equal to zero and attempt to solve: Simplifying d = 0 Solving d = 0 Move all terms containing d to the left, all other terms to the right. Simplifying d = 0Subproblem 2
Set the factor '(-1 + 7d + -1d2)' equal to zero and attempt to solve: Simplifying -1 + 7d + -1d2 = 0 Solving -1 + 7d + -1d2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -7d + d2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -7d + -1 + d2 = 0 + -1 Reorder the terms: 1 + -1 + -7d + d2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -7d + d2 = 0 + -1 -7d + d2 = 0 + -1 Combine like terms: 0 + -1 = -1 -7d + d2 = -1 The d term is -7d. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7d + 12.25 + d2 = -1 + 12.25 Reorder the terms: 12.25 + -7d + d2 = -1 + 12.25 Combine like terms: -1 + 12.25 = 11.25 12.25 + -7d + d2 = 11.25 Factor a perfect square on the left side: (d + -3.5)(d + -3.5) = 11.25 Calculate the square root of the right side: 3.354101966 Break this problem into two subproblems by setting (d + -3.5) equal to 3.354101966 and -3.354101966.Subproblem 1
d + -3.5 = 3.354101966 Simplifying d + -3.5 = 3.354101966 Reorder the terms: -3.5 + d = 3.354101966 Solving -3.5 + d = 3.354101966 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + d = 3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + d = 3.354101966 + 3.5 d = 3.354101966 + 3.5 Combine like terms: 3.354101966 + 3.5 = 6.854101966 d = 6.854101966 Simplifying d = 6.854101966Subproblem 2
d + -3.5 = -3.354101966 Simplifying d + -3.5 = -3.354101966 Reorder the terms: -3.5 + d = -3.354101966 Solving -3.5 + d = -3.354101966 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + d = -3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + d = -3.354101966 + 3.5 d = -3.354101966 + 3.5 Combine like terms: -3.354101966 + 3.5 = 0.145898034 d = 0.145898034 Simplifying d = 0.145898034Solution
The solution to the problem is based on the solutions from the subproblems. d = {6.854101966, 0.145898034}Solution
d = {0, 6.854101966, 0.145898034}
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