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Simplifying 6d2 + -6d + 1 = 0 Reorder the terms: 1 + -6d + 6d2 = 0 Solving 1 + -6d + 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 + -1d + d2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + -1d + -0.1666666667 + d2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + -1d + d2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + -1d + d2 = 0 + -0.1666666667 -1d + d2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 -1d + d2 = -0.1666666667 The d term is -1d. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1d + 0.25 + d2 = -0.1666666667 + 0.25 Reorder the terms: 0.25 + -1d + d2 = -0.1666666667 + 0.25 Combine like terms: -0.1666666667 + 0.25 = 0.0833333333 0.25 + -1d + d2 = 0.0833333333 Factor a perfect square on the left side: (d + -0.5)(d + -0.5) = 0.0833333333 Calculate the square root of the right side: 0.288675135 Break this problem into two subproblems by setting (d + -0.5) equal to 0.288675135 and -0.288675135.Subproblem 1
d + -0.5 = 0.288675135 Simplifying d + -0.5 = 0.288675135 Reorder the terms: -0.5 + d = 0.288675135 Solving -0.5 + d = 0.288675135 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + d = 0.288675135 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = 0.288675135 + 0.5 d = 0.288675135 + 0.5 Combine like terms: 0.288675135 + 0.5 = 0.788675135 d = 0.788675135 Simplifying d = 0.788675135Subproblem 2
d + -0.5 = -0.288675135 Simplifying d + -0.5 = -0.288675135 Reorder the terms: -0.5 + d = -0.288675135 Solving -0.5 + d = -0.288675135 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + d = -0.288675135 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = -0.288675135 + 0.5 d = -0.288675135 + 0.5 Combine like terms: -0.288675135 + 0.5 = 0.211324865 d = 0.211324865 Simplifying d = 0.211324865Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.788675135, 0.211324865}
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