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Simplifying (4d + 3)(d + -1) = (2d + -1) * 2 + 2 Reorder the terms: (3 + 4d)(d + -1) = (2d + -1) * 2 + 2 Reorder the terms: (3 + 4d)(-1 + d) = (2d + -1) * 2 + 2 Multiply (3 + 4d) * (-1 + d) (3(-1 + d) + 4d * (-1 + d)) = (2d + -1) * 2 + 2 ((-1 * 3 + d * 3) + 4d * (-1 + d)) = (2d + -1) * 2 + 2 ((-3 + 3d) + 4d * (-1 + d)) = (2d + -1) * 2 + 2 (-3 + 3d + (-1 * 4d + d * 4d)) = (2d + -1) * 2 + 2 (-3 + 3d + (-4d + 4d2)) = (2d + -1) * 2 + 2 Combine like terms: 3d + -4d = -1d (-3 + -1d + 4d2) = (2d + -1) * 2 + 2 Reorder the terms: -3 + -1d + 4d2 = (-1 + 2d) * 2 + 2 Reorder the terms for easier multiplication: -3 + -1d + 4d2 = 2(-1 + 2d) + 2 -3 + -1d + 4d2 = (-1 * 2 + 2d * 2) + 2 -3 + -1d + 4d2 = (-2 + 4d) + 2 Reorder the terms: -3 + -1d + 4d2 = -2 + 2 + 4d Combine like terms: -2 + 2 = 0 -3 + -1d + 4d2 = 0 + 4d -3 + -1d + 4d2 = 4d Solving -3 + -1d + 4d2 = 4d Solving for variable 'd'. Reorder the terms: -3 + -1d + -4d + 4d2 = 4d + -4d Combine like terms: -1d + -4d = -5d -3 + -5d + 4d2 = 4d + -4d Combine like terms: 4d + -4d = 0 -3 + -5d + 4d2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -1.25d + d2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -1.25d + 0.75 + d2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -1.25d + d2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -1.25d + d2 = 0 + 0.75 -1.25d + d2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -1.25d + d2 = 0.75 The d term is -1.25d. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25d + 0.390625 + d2 = 0.75 + 0.390625 Reorder the terms: 0.390625 + -1.25d + d2 = 0.75 + 0.390625 Combine like terms: 0.75 + 0.390625 = 1.140625 0.390625 + -1.25d + d2 = 1.140625 Factor a perfect square on the left side: (d + -0.625)(d + -0.625) = 1.140625 Calculate the square root of the right side: 1.068000468 Break this problem into two subproblems by setting (d + -0.625) equal to 1.068000468 and -1.068000468.Subproblem 1
d + -0.625 = 1.068000468 Simplifying d + -0.625 = 1.068000468 Reorder the terms: -0.625 + d = 1.068000468 Solving -0.625 + d = 1.068000468 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + d = 1.068000468 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + d = 1.068000468 + 0.625 d = 1.068000468 + 0.625 Combine like terms: 1.068000468 + 0.625 = 1.693000468 d = 1.693000468 Simplifying d = 1.693000468Subproblem 2
d + -0.625 = -1.068000468 Simplifying d + -0.625 = -1.068000468 Reorder the terms: -0.625 + d = -1.068000468 Solving -0.625 + d = -1.068000468 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + d = -1.068000468 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + d = -1.068000468 + 0.625 d = -1.068000468 + 0.625 Combine like terms: -1.068000468 + 0.625 = -0.443000468 d = -0.443000468 Simplifying d = -0.443000468Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.693000468, -0.443000468}
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