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Simplifying 8d3 + -6d2 = d Reorder the terms: -6d2 + 8d3 = d Solving -6d2 + 8d3 = d Solving for variable 'd'. Reorder the terms: -1d + -6d2 + 8d3 = d + -1d Combine like terms: d + -1d = 0 -1d + -6d2 + 8d3 = 0 Factor out the Greatest Common Factor (GCF), 'd'. d(-1 + -6d + 8d2) = 0Subproblem 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 + -6d + 8d2)' equal to zero and attempt to solve: Simplifying -1 + -6d + 8d2 = 0 Solving -1 + -6d + 8d2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + -0.75d + d2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -0.75d + 0.125 + d2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -0.75d + d2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -0.75d + d2 = 0 + 0.125 -0.75d + d2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -0.75d + d2 = 0.125 The d term is -0.75d. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75d + 0.140625 + d2 = 0.125 + 0.140625 Reorder the terms: 0.140625 + -0.75d + d2 = 0.125 + 0.140625 Combine like terms: 0.125 + 0.140625 = 0.265625 0.140625 + -0.75d + d2 = 0.265625 Factor a perfect square on the left side: (d + -0.375)(d + -0.375) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (d + -0.375) equal to 0.515388203 and -0.515388203.Subproblem 1
d + -0.375 = 0.515388203 Simplifying d + -0.375 = 0.515388203 Reorder the terms: -0.375 + d = 0.515388203 Solving -0.375 + d = 0.515388203 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + d = 0.515388203 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + d = 0.515388203 + 0.375 d = 0.515388203 + 0.375 Combine like terms: 0.515388203 + 0.375 = 0.890388203 d = 0.890388203 Simplifying d = 0.890388203Subproblem 2
d + -0.375 = -0.515388203 Simplifying d + -0.375 = -0.515388203 Reorder the terms: -0.375 + d = -0.515388203 Solving -0.375 + d = -0.515388203 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + d = -0.515388203 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + d = -0.515388203 + 0.375 d = -0.515388203 + 0.375 Combine like terms: -0.515388203 + 0.375 = -0.140388203 d = -0.140388203 Simplifying d = -0.140388203Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.890388203, -0.140388203}Solution
d = {0, 0.890388203, -0.140388203}
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