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Simplifying d2 + 6d + -31 = 0 Reorder the terms: -31 + 6d + d2 = 0 Solving -31 + 6d + 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 + 6d + 31 + d2 = 0 + 31 Reorder the terms: -31 + 31 + 6d + d2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + 6d + d2 = 0 + 31 6d + d2 = 0 + 31 Combine like terms: 0 + 31 = 31 6d + d2 = 31 The d term is 6d. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6d + 9 + d2 = 31 + 9 Reorder the terms: 9 + 6d + d2 = 31 + 9 Combine like terms: 31 + 9 = 40 9 + 6d + d2 = 40 Factor a perfect square on the left side: (d + 3)(d + 3) = 40 Calculate the square root of the right side: 6.32455532 Break this problem into two subproblems by setting (d + 3) equal to 6.32455532 and -6.32455532.Subproblem 1
d + 3 = 6.32455532 Simplifying d + 3 = 6.32455532 Reorder the terms: 3 + d = 6.32455532 Solving 3 + d = 6.32455532 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + d = 6.32455532 + -3 Combine like terms: 3 + -3 = 0 0 + d = 6.32455532 + -3 d = 6.32455532 + -3 Combine like terms: 6.32455532 + -3 = 3.32455532 d = 3.32455532 Simplifying d = 3.32455532Subproblem 2
d + 3 = -6.32455532 Simplifying d + 3 = -6.32455532 Reorder the terms: 3 + d = -6.32455532 Solving 3 + d = -6.32455532 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + d = -6.32455532 + -3 Combine like terms: 3 + -3 = 0 0 + d = -6.32455532 + -3 d = -6.32455532 + -3 Combine like terms: -6.32455532 + -3 = -9.32455532 d = -9.32455532 Simplifying d = -9.32455532Solution
The solution to the problem is based on the solutions from the subproblems. d = {3.32455532, -9.32455532}
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