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Simplifying 2d2 + 9d + 8 = 0 Reorder the terms: 8 + 9d + 2d2 = 0 Solving 8 + 9d + 2d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 4 + 4.5d + d2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + 4.5d + -4 + d2 = 0 + -4 Reorder the terms: 4 + -4 + 4.5d + d2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + 4.5d + d2 = 0 + -4 4.5d + d2 = 0 + -4 Combine like terms: 0 + -4 = -4 4.5d + d2 = -4 The d term is 4.5d. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5d + 5.0625 + d2 = -4 + 5.0625 Reorder the terms: 5.0625 + 4.5d + d2 = -4 + 5.0625 Combine like terms: -4 + 5.0625 = 1.0625 5.0625 + 4.5d + d2 = 1.0625 Factor a perfect square on the left side: (d + 2.25)(d + 2.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (d + 2.25) equal to 1.030776406 and -1.030776406.Subproblem 1
d + 2.25 = 1.030776406 Simplifying d + 2.25 = 1.030776406 Reorder the terms: 2.25 + d = 1.030776406 Solving 2.25 + d = 1.030776406 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + d = 1.030776406 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + d = 1.030776406 + -2.25 d = 1.030776406 + -2.25 Combine like terms: 1.030776406 + -2.25 = -1.219223594 d = -1.219223594 Simplifying d = -1.219223594Subproblem 2
d + 2.25 = -1.030776406 Simplifying d + 2.25 = -1.030776406 Reorder the terms: 2.25 + d = -1.030776406 Solving 2.25 + d = -1.030776406 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + d = -1.030776406 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + d = -1.030776406 + -2.25 d = -1.030776406 + -2.25 Combine like terms: -1.030776406 + -2.25 = -3.280776406 d = -3.280776406 Simplifying d = -3.280776406Solution
The solution to the problem is based on the solutions from the subproblems. d = {-1.219223594, -3.280776406}
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