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Simplifying 16d(-4 + -5d) = -67 (-4 * 16d + -5d * 16d) = -67 (-64d + -80d2) = -67 Solving -64d + -80d2 = -67 Solving for variable 'd'. Reorder the terms: 67 + -64d + -80d2 = -67 + 67 Combine like terms: -67 + 67 = 0 67 + -64d + -80d2 = 0 Begin completing the square. Divide all terms by -80 the coefficient of the squared term: Divide each side by '-80'. -0.8375 + 0.8d + d2 = 0.0 Move the constant term to the right: Add '0.8375' to each side of the equation. -0.8375 + 0.8d + 0.8375 + d2 = 0.0 + 0.8375 Reorder the terms: -0.8375 + 0.8375 + 0.8d + d2 = 0.0 + 0.8375 Combine like terms: -0.8375 + 0.8375 = 0.0000 0.0000 + 0.8d + d2 = 0.0 + 0.8375 0.8d + d2 = 0.0 + 0.8375 Combine like terms: 0.0 + 0.8375 = 0.8375 0.8d + d2 = 0.8375 The d term is 0.8d. Take half its coefficient (0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. 0.8d + 0.16 + d2 = 0.8375 + 0.16 Reorder the terms: 0.16 + 0.8d + d2 = 0.8375 + 0.16 Combine like terms: 0.8375 + 0.16 = 0.9975 0.16 + 0.8d + d2 = 0.9975 Factor a perfect square on the left side: (d + 0.4)(d + 0.4) = 0.9975 Calculate the square root of the right side: 0.998749218 Break this problem into two subproblems by setting (d + 0.4) equal to 0.998749218 and -0.998749218.Subproblem 1
d + 0.4 = 0.998749218 Simplifying d + 0.4 = 0.998749218 Reorder the terms: 0.4 + d = 0.998749218 Solving 0.4 + d = 0.998749218 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + d = 0.998749218 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + d = 0.998749218 + -0.4 d = 0.998749218 + -0.4 Combine like terms: 0.998749218 + -0.4 = 0.598749218 d = 0.598749218 Simplifying d = 0.598749218Subproblem 2
d + 0.4 = -0.998749218 Simplifying d + 0.4 = -0.998749218 Reorder the terms: 0.4 + d = -0.998749218 Solving 0.4 + d = -0.998749218 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + d = -0.998749218 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + d = -0.998749218 + -0.4 d = -0.998749218 + -0.4 Combine like terms: -0.998749218 + -0.4 = -1.398749218 d = -1.398749218 Simplifying d = -1.398749218Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.598749218, -1.398749218}
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