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Simplifying 100 = 2d2 + 6d + 9 Reorder the terms: 100 = 9 + 6d + 2d2 Solving 100 = 9 + 6d + 2d2 Solving for variable 'd'. Combine like terms: 100 + -9 = 91 91 + -6d + -2d2 = 9 + 6d + 2d2 + -9 + -6d + -2d2 Reorder the terms: 91 + -6d + -2d2 = 9 + -9 + 6d + -6d + 2d2 + -2d2 Combine like terms: 9 + -9 = 0 91 + -6d + -2d2 = 0 + 6d + -6d + 2d2 + -2d2 91 + -6d + -2d2 = 6d + -6d + 2d2 + -2d2 Combine like terms: 6d + -6d = 0 91 + -6d + -2d2 = 0 + 2d2 + -2d2 91 + -6d + -2d2 = 2d2 + -2d2 Combine like terms: 2d2 + -2d2 = 0 91 + -6d + -2d2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -45.5 + 3d + d2 = 0 Move the constant term to the right: Add '45.5' to each side of the equation. -45.5 + 3d + 45.5 + d2 = 0 + 45.5 Reorder the terms: -45.5 + 45.5 + 3d + d2 = 0 + 45.5 Combine like terms: -45.5 + 45.5 = 0.0 0.0 + 3d + d2 = 0 + 45.5 3d + d2 = 0 + 45.5 Combine like terms: 0 + 45.5 = 45.5 3d + d2 = 45.5 The d term is 3d. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3d + 2.25 + d2 = 45.5 + 2.25 Reorder the terms: 2.25 + 3d + d2 = 45.5 + 2.25 Combine like terms: 45.5 + 2.25 = 47.75 2.25 + 3d + d2 = 47.75 Factor a perfect square on the left side: (d + 1.5)(d + 1.5) = 47.75 Calculate the square root of the right side: 6.910137481 Break this problem into two subproblems by setting (d + 1.5) equal to 6.910137481 and -6.910137481.Subproblem 1
d + 1.5 = 6.910137481 Simplifying d + 1.5 = 6.910137481 Reorder the terms: 1.5 + d = 6.910137481 Solving 1.5 + d = 6.910137481 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + d = 6.910137481 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + d = 6.910137481 + -1.5 d = 6.910137481 + -1.5 Combine like terms: 6.910137481 + -1.5 = 5.410137481 d = 5.410137481 Simplifying d = 5.410137481Subproblem 2
d + 1.5 = -6.910137481 Simplifying d + 1.5 = -6.910137481 Reorder the terms: 1.5 + d = -6.910137481 Solving 1.5 + d = -6.910137481 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + d = -6.910137481 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + d = -6.910137481 + -1.5 d = -6.910137481 + -1.5 Combine like terms: -6.910137481 + -1.5 = -8.410137481 d = -8.410137481 Simplifying d = -8.410137481Solution
The solution to the problem is based on the solutions from the subproblems. d = {5.410137481, -8.410137481}
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