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Simplifying d(8d + -8) = -1 Reorder the terms: d(-8 + 8d) = -1 (-8 * d + 8d * d) = -1 (-8d + 8d2) = -1 Solving -8d + 8d2 = -1 Solving for variable 'd'. Reorder the terms: 1 + -8d + 8d2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + -8d + 8d2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.125 + -1d + d2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -1d + -0.125 + d2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -1d + d2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -1d + d2 = 0 + -0.125 -1d + d2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -1d + d2 = -0.125 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 = -0.125 + 0.25 Reorder the terms: 0.25 + -1d + d2 = -0.125 + 0.25 Combine like terms: -0.125 + 0.25 = 0.125 0.25 + -1d + d2 = 0.125 Factor a perfect square on the left side: (d + -0.5)(d + -0.5) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (d + -0.5) equal to 0.353553391 and -0.353553391.Subproblem 1
d + -0.5 = 0.353553391 Simplifying d + -0.5 = 0.353553391 Reorder the terms: -0.5 + d = 0.353553391 Solving -0.5 + d = 0.353553391 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 = 0.353553391 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = 0.353553391 + 0.5 d = 0.353553391 + 0.5 Combine like terms: 0.353553391 + 0.5 = 0.853553391 d = 0.853553391 Simplifying d = 0.853553391Subproblem 2
d + -0.5 = -0.353553391 Simplifying d + -0.5 = -0.353553391 Reorder the terms: -0.5 + d = -0.353553391 Solving -0.5 + d = -0.353553391 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 = -0.353553391 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = -0.353553391 + 0.5 d = -0.353553391 + 0.5 Combine like terms: -0.353553391 + 0.5 = 0.146446609 d = 0.146446609 Simplifying d = 0.146446609Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.853553391, 0.146446609}
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