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Simplifying 8 + -1d(d + 4) = 3d(d + 2) + 5 Reorder the terms: 8 + -1d(4 + d) = 3d(d + 2) + 5 8 + (4 * -1d + d * -1d) = 3d(d + 2) + 5 8 + (-4d + -1d2) = 3d(d + 2) + 5 Reorder the terms: 8 + -4d + -1d2 = 3d(2 + d) + 5 8 + -4d + -1d2 = (2 * 3d + d * 3d) + 5 8 + -4d + -1d2 = (6d + 3d2) + 5 Reorder the terms: 8 + -4d + -1d2 = 5 + 6d + 3d2 Solving 8 + -4d + -1d2 = 5 + 6d + 3d2 Solving for variable 'd'. Reorder the terms: 8 + -5 + -4d + -6d + -1d2 + -3d2 = 5 + 6d + 3d2 + -5 + -6d + -3d2 Combine like terms: 8 + -5 = 3 3 + -4d + -6d + -1d2 + -3d2 = 5 + 6d + 3d2 + -5 + -6d + -3d2 Combine like terms: -4d + -6d = -10d 3 + -10d + -1d2 + -3d2 = 5 + 6d + 3d2 + -5 + -6d + -3d2 Combine like terms: -1d2 + -3d2 = -4d2 3 + -10d + -4d2 = 5 + 6d + 3d2 + -5 + -6d + -3d2 Reorder the terms: 3 + -10d + -4d2 = 5 + -5 + 6d + -6d + 3d2 + -3d2 Combine like terms: 5 + -5 = 0 3 + -10d + -4d2 = 0 + 6d + -6d + 3d2 + -3d2 3 + -10d + -4d2 = 6d + -6d + 3d2 + -3d2 Combine like terms: 6d + -6d = 0 3 + -10d + -4d2 = 0 + 3d2 + -3d2 3 + -10d + -4d2 = 3d2 + -3d2 Combine like terms: 3d2 + -3d2 = 0 3 + -10d + -4d2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.75 + 2.5d + d2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 2.5d + 0.75 + d2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 2.5d + d2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 2.5d + d2 = 0 + 0.75 2.5d + d2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 2.5d + d2 = 0.75 The d term is 2.5d. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5d + 1.5625 + d2 = 0.75 + 1.5625 Reorder the terms: 1.5625 + 2.5d + d2 = 0.75 + 1.5625 Combine like terms: 0.75 + 1.5625 = 2.3125 1.5625 + 2.5d + d2 = 2.3125 Factor a perfect square on the left side: (d + 1.25)(d + 1.25) = 2.3125 Calculate the square root of the right side: 1.520690633 Break this problem into two subproblems by setting (d + 1.25) equal to 1.520690633 and -1.520690633.Subproblem 1
d + 1.25 = 1.520690633 Simplifying d + 1.25 = 1.520690633 Reorder the terms: 1.25 + d = 1.520690633 Solving 1.25 + d = 1.520690633 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + d = 1.520690633 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + d = 1.520690633 + -1.25 d = 1.520690633 + -1.25 Combine like terms: 1.520690633 + -1.25 = 0.270690633 d = 0.270690633 Simplifying d = 0.270690633Subproblem 2
d + 1.25 = -1.520690633 Simplifying d + 1.25 = -1.520690633 Reorder the terms: 1.25 + d = -1.520690633 Solving 1.25 + d = -1.520690633 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + d = -1.520690633 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + d = -1.520690633 + -1.25 d = -1.520690633 + -1.25 Combine like terms: -1.520690633 + -1.25 = -2.770690633 d = -2.770690633 Simplifying d = -2.770690633Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.270690633, -2.770690633}
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