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Simplifying d2 + 12d + -31 = 0 Reorder the terms: -31 + 12d + d2 = 0 Solving -31 + 12d + d2 = 0 Solving for variable 'd'. Begin completing the square. Move the constant term to the right: Add '31' to each side of the equation. -31 + 12d + 31 + d2 = 0 + 31 Reorder the terms: -31 + 31 + 12d + d2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + 12d + d2 = 0 + 31 12d + d2 = 0 + 31 Combine like terms: 0 + 31 = 31 12d + d2 = 31 The d term is 12d. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12d + 36 + d2 = 31 + 36 Reorder the terms: 36 + 12d + d2 = 31 + 36 Combine like terms: 31 + 36 = 67 36 + 12d + d2 = 67 Factor a perfect square on the left side: (d + 6)(d + 6) = 67 Calculate the square root of the right side: 8.185352772 Break this problem into two subproblems by setting (d + 6) equal to 8.185352772 and -8.185352772.Subproblem 1
d + 6 = 8.185352772 Simplifying d + 6 = 8.185352772 Reorder the terms: 6 + d = 8.185352772 Solving 6 + d = 8.185352772 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + d = 8.185352772 + -6 Combine like terms: 6 + -6 = 0 0 + d = 8.185352772 + -6 d = 8.185352772 + -6 Combine like terms: 8.185352772 + -6 = 2.185352772 d = 2.185352772 Simplifying d = 2.185352772Subproblem 2
d + 6 = -8.185352772 Simplifying d + 6 = -8.185352772 Reorder the terms: 6 + d = -8.185352772 Solving 6 + d = -8.185352772 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + d = -8.185352772 + -6 Combine like terms: 6 + -6 = 0 0 + d = -8.185352772 + -6 d = -8.185352772 + -6 Combine like terms: -8.185352772 + -6 = -14.185352772 d = -14.185352772 Simplifying d = -14.185352772Solution
The solution to the problem is based on the solutions from the subproblems. d = {2.185352772, -14.185352772}
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