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Simplifying 8d2 + 9d + -1 = 0 Reorder the terms: -1 + 9d + 8d2 = 0 Solving -1 + 9d + 8d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + 1.125d + d2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 1.125d + 0.125 + d2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 1.125d + d2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 1.125d + d2 = 0 + 0.125 1.125d + d2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 1.125d + d2 = 0.125 The d term is 1.125d. Take half its coefficient (0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. 1.125d + 0.31640625 + d2 = 0.125 + 0.31640625 Reorder the terms: 0.31640625 + 1.125d + d2 = 0.125 + 0.31640625 Combine like terms: 0.125 + 0.31640625 = 0.44140625 0.31640625 + 1.125d + d2 = 0.44140625 Factor a perfect square on the left side: (d + 0.5625)(d + 0.5625) = 0.44140625 Calculate the square root of the right side: 0.664384113 Break this problem into two subproblems by setting (d + 0.5625) equal to 0.664384113 and -0.664384113.Subproblem 1
d + 0.5625 = 0.664384113 Simplifying d + 0.5625 = 0.664384113 Reorder the terms: 0.5625 + d = 0.664384113 Solving 0.5625 + d = 0.664384113 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + d = 0.664384113 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + d = 0.664384113 + -0.5625 d = 0.664384113 + -0.5625 Combine like terms: 0.664384113 + -0.5625 = 0.101884113 d = 0.101884113 Simplifying d = 0.101884113Subproblem 2
d + 0.5625 = -0.664384113 Simplifying d + 0.5625 = -0.664384113 Reorder the terms: 0.5625 + d = -0.664384113 Solving 0.5625 + d = -0.664384113 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + d = -0.664384113 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + d = -0.664384113 + -0.5625 d = -0.664384113 + -0.5625 Combine like terms: -0.664384113 + -0.5625 = -1.226884113 d = -1.226884113 Simplifying d = -1.226884113Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.101884113, -1.226884113}
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