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Simplifying d(d + 1) = 2(d + 4) Reorder the terms: d(1 + d) = 2(d + 4) (1 * d + d * d) = 2(d + 4) (1d + d2) = 2(d + 4) Reorder the terms: 1d + d2 = 2(4 + d) 1d + d2 = (4 * 2 + d * 2) 1d + d2 = (8 + 2d) Solving 1d + d2 = 8 + 2d Solving for variable 'd'. Reorder the terms: -8 + 1d + -2d + d2 = 8 + 2d + -8 + -2d Combine like terms: 1d + -2d = -1d -8 + -1d + d2 = 8 + 2d + -8 + -2d Reorder the terms: -8 + -1d + d2 = 8 + -8 + 2d + -2d Combine like terms: 8 + -8 = 0 -8 + -1d + d2 = 0 + 2d + -2d -8 + -1d + d2 = 2d + -2d Combine like terms: 2d + -2d = 0 -8 + -1d + d2 = 0 Begin completing the square. Move the constant term to the right: Add '8' to each side of the equation. -8 + -1d + 8 + d2 = 0 + 8 Reorder the terms: -8 + 8 + -1d + d2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -1d + d2 = 0 + 8 -1d + d2 = 0 + 8 Combine like terms: 0 + 8 = 8 -1d + d2 = 8 The d term is -1d. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1d + 0.25 + d2 = 8 + 0.25 Reorder the terms: 0.25 + -1d + d2 = 8 + 0.25 Combine like terms: 8 + 0.25 = 8.25 0.25 + -1d + d2 = 8.25 Factor a perfect square on the left side: (d + -0.5)(d + -0.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (d + -0.5) equal to 2.872281323 and -2.872281323.Subproblem 1
d + -0.5 = 2.872281323 Simplifying d + -0.5 = 2.872281323 Reorder the terms: -0.5 + d = 2.872281323 Solving -0.5 + d = 2.872281323 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + d = 2.872281323 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = 2.872281323 + 0.5 d = 2.872281323 + 0.5 Combine like terms: 2.872281323 + 0.5 = 3.372281323 d = 3.372281323 Simplifying d = 3.372281323Subproblem 2
d + -0.5 = -2.872281323 Simplifying d + -0.5 = -2.872281323 Reorder the terms: -0.5 + d = -2.872281323 Solving -0.5 + d = -2.872281323 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + d = -2.872281323 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = -2.872281323 + 0.5 d = -2.872281323 + 0.5 Combine like terms: -2.872281323 + 0.5 = -2.372281323 d = -2.372281323 Simplifying d = -2.372281323Solution
The solution to the problem is based on the solutions from the subproblems. d = {3.372281323, -2.372281323}
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