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Simplifying 3d2 + -8d + 3 = 0 Reorder the terms: 3 + -8d + 3d2 = 0 Solving 3 + -8d + 3d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1 + -2.666666667d + d2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.666666667d + -1 + d2 = 0 + -1 Reorder the terms: 1 + -1 + -2.666666667d + d2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.666666667d + d2 = 0 + -1 -2.666666667d + d2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.666666667d + d2 = -1 The d term is -2.666666667d. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667d + 1.777777780 + d2 = -1 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667d + d2 = -1 + 1.777777780 Combine like terms: -1 + 1.777777780 = 0.77777778 1.777777780 + -2.666666667d + d2 = 0.77777778 Factor a perfect square on the left side: (d + -1.333333334)(d + -1.333333334) = 0.77777778 Calculate the square root of the right side: 0.881917105 Break this problem into two subproblems by setting (d + -1.333333334) equal to 0.881917105 and -0.881917105.Subproblem 1
d + -1.333333334 = 0.881917105 Simplifying d + -1.333333334 = 0.881917105 Reorder the terms: -1.333333334 + d = 0.881917105 Solving -1.333333334 + d = 0.881917105 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + d = 0.881917105 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + d = 0.881917105 + 1.333333334 d = 0.881917105 + 1.333333334 Combine like terms: 0.881917105 + 1.333333334 = 2.215250439 d = 2.215250439 Simplifying d = 2.215250439Subproblem 2
d + -1.333333334 = -0.881917105 Simplifying d + -1.333333334 = -0.881917105 Reorder the terms: -1.333333334 + d = -0.881917105 Solving -1.333333334 + d = -0.881917105 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + d = -0.881917105 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + d = -0.881917105 + 1.333333334 d = -0.881917105 + 1.333333334 Combine like terms: -0.881917105 + 1.333333334 = 0.451416229 d = 0.451416229 Simplifying d = 0.451416229Solution
The solution to the problem is based on the solutions from the subproblems. d = {2.215250439, 0.451416229}
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