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Simplifying 12d2 + 8d + -6 = 0 Reorder the terms: -6 + 8d + 12d2 = 0 Solving -6 + 8d + 12d2 = 0 Solving for variable 'd'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 4d + 6d2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 4d + 6d2)' equal to zero and attempt to solve: Simplifying -3 + 4d + 6d2 = 0 Solving -3 + 4d + 6d2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + 0.6666666667d + d2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.6666666667d + 0.5 + d2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.6666666667d + d2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.6666666667d + d2 = 0 + 0.5 0.6666666667d + d2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.6666666667d + d2 = 0.5 The d term is 0.6666666667d. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667d + 0.1111111112 + d2 = 0.5 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667d + d2 = 0.5 + 0.1111111112 Combine like terms: 0.5 + 0.1111111112 = 0.6111111112 0.1111111112 + 0.6666666667d + d2 = 0.6111111112 Factor a perfect square on the left side: (d + 0.3333333334)(d + 0.3333333334) = 0.6111111112 Calculate the square root of the right side: 0.78173596 Break this problem into two subproblems by setting (d + 0.3333333334) equal to 0.78173596 and -0.78173596.Subproblem 1
d + 0.3333333334 = 0.78173596 Simplifying d + 0.3333333334 = 0.78173596 Reorder the terms: 0.3333333334 + d = 0.78173596 Solving 0.3333333334 + d = 0.78173596 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + d = 0.78173596 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + d = 0.78173596 + -0.3333333334 d = 0.78173596 + -0.3333333334 Combine like terms: 0.78173596 + -0.3333333334 = 0.4484026266 d = 0.4484026266 Simplifying d = 0.4484026266Subproblem 2
d + 0.3333333334 = -0.78173596 Simplifying d + 0.3333333334 = -0.78173596 Reorder the terms: 0.3333333334 + d = -0.78173596 Solving 0.3333333334 + d = -0.78173596 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + d = -0.78173596 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + d = -0.78173596 + -0.3333333334 d = -0.78173596 + -0.3333333334 Combine like terms: -0.78173596 + -0.3333333334 = -1.1150692934 d = -1.1150692934 Simplifying d = -1.1150692934Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.4484026266, -1.1150692934}Solution
d = {0.4484026266, -1.1150692934}
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