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Simplifying (0.7d + 1)(1.5d + -2) = (2.5 + d)(4 + -1.3d) Reorder the terms: (1 + 0.7d)(1.5d + -2) = (2.5 + d)(4 + -1.3d) Reorder the terms: (1 + 0.7d)(-2 + 1.5d) = (2.5 + d)(4 + -1.3d) Multiply (1 + 0.7d) * (-2 + 1.5d) (1(-2 + 1.5d) + 0.7d * (-2 + 1.5d)) = (2.5 + d)(4 + -1.3d) ((-2 * 1 + 1.5d * 1) + 0.7d * (-2 + 1.5d)) = (2.5 + d)(4 + -1.3d) ((-2 + 1.5d) + 0.7d * (-2 + 1.5d)) = (2.5 + d)(4 + -1.3d) (-2 + 1.5d + (-2 * 0.7d + 1.5d * 0.7d)) = (2.5 + d)(4 + -1.3d) (-2 + 1.5d + (-1.4d + 1.05d2)) = (2.5 + d)(4 + -1.3d) Combine like terms: 1.5d + -1.4d = 0.1d (-2 + 0.1d + 1.05d2) = (2.5 + d)(4 + -1.3d) Multiply (2.5 + d) * (4 + -1.3d) -2 + 0.1d + 1.05d2 = (2.5(4 + -1.3d) + d(4 + -1.3d)) -2 + 0.1d + 1.05d2 = ((4 * 2.5 + -1.3d * 2.5) + d(4 + -1.3d)) -2 + 0.1d + 1.05d2 = ((10 + -3.25d) + d(4 + -1.3d)) -2 + 0.1d + 1.05d2 = (10 + -3.25d + (4 * d + -1.3d * d)) -2 + 0.1d + 1.05d2 = (10 + -3.25d + (4d + -1.3d2)) Combine like terms: -3.25d + 4d = 0.75d -2 + 0.1d + 1.05d2 = (10 + 0.75d + -1.3d2) Solving -2 + 0.1d + 1.05d2 = 10 + 0.75d + -1.3d2 Solving for variable 'd'. Reorder the terms: -2 + -10 + 0.1d + -0.75d + 1.05d2 + 1.3d2 = 10 + 0.75d + -1.3d2 + -10 + -0.75d + 1.3d2 Combine like terms: -2 + -10 = -12 -12 + 0.1d + -0.75d + 1.05d2 + 1.3d2 = 10 + 0.75d + -1.3d2 + -10 + -0.75d + 1.3d2 Combine like terms: 0.1d + -0.75d = -0.65d -12 + -0.65d + 1.05d2 + 1.3d2 = 10 + 0.75d + -1.3d2 + -10 + -0.75d + 1.3d2 Combine like terms: 1.05d2 + 1.3d2 = 2.35d2 -12 + -0.65d + 2.35d2 = 10 + 0.75d + -1.3d2 + -10 + -0.75d + 1.3d2 Reorder the terms: -12 + -0.65d + 2.35d2 = 10 + -10 + 0.75d + -0.75d + -1.3d2 + 1.3d2 Combine like terms: 10 + -10 = 0 -12 + -0.65d + 2.35d2 = 0 + 0.75d + -0.75d + -1.3d2 + 1.3d2 -12 + -0.65d + 2.35d2 = 0.75d + -0.75d + -1.3d2 + 1.3d2 Combine like terms: 0.75d + -0.75d = 0.00 -12 + -0.65d + 2.35d2 = 0.00 + -1.3d2 + 1.3d2 -12 + -0.65d + 2.35d2 = -1.3d2 + 1.3d2 Combine like terms: -1.3d2 + 1.3d2 = 0.0 -12 + -0.65d + 2.35d2 = 0.0 Begin completing the square. Divide all terms by 2.35 the coefficient of the squared term: Divide each side by '2.35'. -5.106382979 + -0.2765957447d + d2 = 0 Move the constant term to the right: Add '5.106382979' to each side of the equation. -5.106382979 + -0.2765957447d + 5.106382979 + d2 = 0 + 5.106382979 Reorder the terms: -5.106382979 + 5.106382979 + -0.2765957447d + d2 = 0 + 5.106382979 Combine like terms: -5.106382979 + 5.106382979 = 0.000000000 0.000000000 + -0.2765957447d + d2 = 0 + 5.106382979 -0.2765957447d + d2 = 0 + 5.106382979 Combine like terms: 0 + 5.106382979 = 5.106382979 -0.2765957447d + d2 = 5.106382979 The d term is -0.2765957447d. Take half its coefficient (-0.1382978724). Square it (0.01912630151) and add it to both sides. Add '0.01912630151' to each side of the equation. -0.2765957447d + 0.01912630151 + d2 = 5.106382979 + 0.01912630151 Reorder the terms: 0.01912630151 + -0.2765957447d + d2 = 5.106382979 + 0.01912630151 Combine like terms: 5.106382979 + 0.01912630151 = 5.12550928051 0.01912630151 + -0.2765957447d + d2 = 5.12550928051 Factor a perfect square on the left side: (d + -0.1382978724)(d + -0.1382978724) = 5.12550928051 Calculate the square root of the right side: 2.263958763 Break this problem into two subproblems by setting (d + -0.1382978724) equal to 2.263958763 and -2.263958763.Subproblem 1
d + -0.1382978724 = 2.263958763 Simplifying d + -0.1382978724 = 2.263958763 Reorder the terms: -0.1382978724 + d = 2.263958763 Solving -0.1382978724 + d = 2.263958763 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.1382978724' to each side of the equation. -0.1382978724 + 0.1382978724 + d = 2.263958763 + 0.1382978724 Combine like terms: -0.1382978724 + 0.1382978724 = 0.0000000000 0.0000000000 + d = 2.263958763 + 0.1382978724 d = 2.263958763 + 0.1382978724 Combine like terms: 2.263958763 + 0.1382978724 = 2.4022566354 d = 2.4022566354 Simplifying d = 2.4022566354Subproblem 2
d + -0.1382978724 = -2.263958763 Simplifying d + -0.1382978724 = -2.263958763 Reorder the terms: -0.1382978724 + d = -2.263958763 Solving -0.1382978724 + d = -2.263958763 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.1382978724' to each side of the equation. -0.1382978724 + 0.1382978724 + d = -2.263958763 + 0.1382978724 Combine like terms: -0.1382978724 + 0.1382978724 = 0.0000000000 0.0000000000 + d = -2.263958763 + 0.1382978724 d = -2.263958763 + 0.1382978724 Combine like terms: -2.263958763 + 0.1382978724 = -2.1256608906 d = -2.1256608906 Simplifying d = -2.1256608906Solution
The solution to the problem is based on the solutions from the subproblems. d = {2.4022566354, -2.1256608906}
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