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Simplifying 3.14d2 + 7 * 3.14d + 12 * 3.14 = 0 Multiply 7 * 3.14 3.14d2 + 21.98d + 12 * 3.14 = 0 Multiply 12 * 3.14 3.14d2 + 21.98d + 37.68 = 0 Reorder the terms: 37.68 + 21.98d + 3.14d2 = 0 Solving 37.68 + 21.98d + 3.14d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 3.14 the coefficient of the squared term: Divide each side by '3.14'. 12 + 7d + d2 = 0 Move the constant term to the right: Add '-12' to each side of the equation. 12 + 7d + -12 + d2 = 0 + -12 Reorder the terms: 12 + -12 + 7d + d2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + 7d + d2 = 0 + -12 7d + d2 = 0 + -12 Combine like terms: 0 + -12 = -12 7d + d2 = -12 The d term is 7d. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7d + 12.25 + d2 = -12 + 12.25 Reorder the terms: 12.25 + 7d + d2 = -12 + 12.25 Combine like terms: -12 + 12.25 = 0.25 12.25 + 7d + d2 = 0.25 Factor a perfect square on the left side: (d + 3.5)(d + 3.5) = 0.25 Calculate the square root of the right side: 0.5 Break this problem into two subproblems by setting (d + 3.5) equal to 0.5 and -0.5.Subproblem 1
d + 3.5 = 0.5 Simplifying d + 3.5 = 0.5 Reorder the terms: 3.5 + d = 0.5 Solving 3.5 + d = 0.5 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + d = 0.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + d = 0.5 + -3.5 d = 0.5 + -3.5 Combine like terms: 0.5 + -3.5 = -3 d = -3 Simplifying d = -3Subproblem 2
d + 3.5 = -0.5 Simplifying d + 3.5 = -0.5 Reorder the terms: 3.5 + d = -0.5 Solving 3.5 + d = -0.5 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + d = -0.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + d = -0.5 + -3.5 d = -0.5 + -3.5 Combine like terms: -0.5 + -3.5 = -4 d = -4 Simplifying d = -4Solution
The solution to the problem is based on the solutions from the subproblems. d = {-3, -4}
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