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Simplifying 23d + -50 + -1d2 = 0 Reorder the terms: -50 + 23d + -1d2 = 0 Solving -50 + 23d + -1d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 50 + -23d + d2 = 0 Move the constant term to the right: Add '-50' to each side of the equation. 50 + -23d + -50 + d2 = 0 + -50 Reorder the terms: 50 + -50 + -23d + d2 = 0 + -50 Combine like terms: 50 + -50 = 0 0 + -23d + d2 = 0 + -50 -23d + d2 = 0 + -50 Combine like terms: 0 + -50 = -50 -23d + d2 = -50 The d term is -23d. Take half its coefficient (-11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. -23d + 132.25 + d2 = -50 + 132.25 Reorder the terms: 132.25 + -23d + d2 = -50 + 132.25 Combine like terms: -50 + 132.25 = 82.25 132.25 + -23d + d2 = 82.25 Factor a perfect square on the left side: (d + -11.5)(d + -11.5) = 82.25 Calculate the square root of the right side: 9.069178574 Break this problem into two subproblems by setting (d + -11.5) equal to 9.069178574 and -9.069178574.Subproblem 1
d + -11.5 = 9.069178574 Simplifying d + -11.5 = 9.069178574 Reorder the terms: -11.5 + d = 9.069178574 Solving -11.5 + d = 9.069178574 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + d = 9.069178574 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + d = 9.069178574 + 11.5 d = 9.069178574 + 11.5 Combine like terms: 9.069178574 + 11.5 = 20.569178574 d = 20.569178574 Simplifying d = 20.569178574Subproblem 2
d + -11.5 = -9.069178574 Simplifying d + -11.5 = -9.069178574 Reorder the terms: -11.5 + d = -9.069178574 Solving -11.5 + d = -9.069178574 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + d = -9.069178574 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + d = -9.069178574 + 11.5 d = -9.069178574 + 11.5 Combine like terms: -9.069178574 + 11.5 = 2.430821426 d = 2.430821426 Simplifying d = 2.430821426Solution
The solution to the problem is based on the solutions from the subproblems. d = {20.569178574, 2.430821426}
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