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Simplifying 5000d2 + 15d + -16 = 0 Reorder the terms: -16 + 15d + 5000d2 = 0 Solving -16 + 15d + 5000d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 5000 the coefficient of the squared term: Divide each side by '5000'. -0.0032 + 0.003d + d2 = 0 Move the constant term to the right: Add '0.0032' to each side of the equation. -0.0032 + 0.003d + 0.0032 + d2 = 0 + 0.0032 Reorder the terms: -0.0032 + 0.0032 + 0.003d + d2 = 0 + 0.0032 Combine like terms: -0.0032 + 0.0032 = 0.0000 0.0000 + 0.003d + d2 = 0 + 0.0032 0.003d + d2 = 0 + 0.0032 Combine like terms: 0 + 0.0032 = 0.0032 0.003d + d2 = 0.0032 The d term is 0.003d. Take half its coefficient (0.0015). Square it (0.00000225) and add it to both sides. Add '0.00000225' to each side of the equation. 0.003d + 0.00000225 + d2 = 0.0032 + 0.00000225 Reorder the terms: 0.00000225 + 0.003d + d2 = 0.0032 + 0.00000225 Combine like terms: 0.0032 + 0.00000225 = 0.00320225 0.00000225 + 0.003d + d2 = 0.00320225 Factor a perfect square on the left side: (d + 0.0015)(d + 0.0015) = 0.00320225 Calculate the square root of the right side: 0.056588426 Break this problem into two subproblems by setting (d + 0.0015) equal to 0.056588426 and -0.056588426.Subproblem 1
d + 0.0015 = 0.056588426 Simplifying d + 0.0015 = 0.056588426 Reorder the terms: 0.0015 + d = 0.056588426 Solving 0.0015 + d = 0.056588426 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.0015' to each side of the equation. 0.0015 + -0.0015 + d = 0.056588426 + -0.0015 Combine like terms: 0.0015 + -0.0015 = 0.0000 0.0000 + d = 0.056588426 + -0.0015 d = 0.056588426 + -0.0015 Combine like terms: 0.056588426 + -0.0015 = 0.055088426 d = 0.055088426 Simplifying d = 0.055088426Subproblem 2
d + 0.0015 = -0.056588426 Simplifying d + 0.0015 = -0.056588426 Reorder the terms: 0.0015 + d = -0.056588426 Solving 0.0015 + d = -0.056588426 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.0015' to each side of the equation. 0.0015 + -0.0015 + d = -0.056588426 + -0.0015 Combine like terms: 0.0015 + -0.0015 = 0.0000 0.0000 + d = -0.056588426 + -0.0015 d = -0.056588426 + -0.0015 Combine like terms: -0.056588426 + -0.0015 = -0.058088426 d = -0.058088426 Simplifying d = -0.058088426Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.055088426, -0.058088426}
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