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Simplifying (d + 0.5)(d + 0.5) = 1 Reorder the terms: (0.5 + d)(d + 0.5) = 1 Reorder the terms: (0.5 + d)(0.5 + d) = 1 Multiply (0.5 + d) * (0.5 + d) (0.5(0.5 + d) + d(0.5 + d)) = 1 ((0.5 * 0.5 + d * 0.5) + d(0.5 + d)) = 1 ((0.25 + 0.5d) + d(0.5 + d)) = 1 (0.25 + 0.5d + (0.5 * d + d * d)) = 1 (0.25 + 0.5d + (0.5d + d2)) = 1 Combine like terms: 0.5d + 0.5d = 1d (0.25 + 1d + d2) = 1 Solving 0.25 + 1d + d2 = 1 Solving for variable 'd'. Reorder the terms: 0.25 + -1 + 1d + d2 = 1 + -1 Combine like terms: 0.25 + -1 = -0.75 -0.75 + 1d + d2 = 1 + -1 Combine like terms: 1 + -1 = 0 -0.75 + 1d + d2 = 0 Begin completing the square. Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1d + 0.75 + d2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1d + d2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1d + d2 = 0 + 0.75 1d + d2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1d + d2 = 0.75 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. d + 0.25 + d2 = 0.75 + 0.25 Reorder the terms: 0.25 + d + d2 = 0.75 + 0.25 Combine like terms: 0.75 + 0.25 = 1 0.25 + d + d2 = 1 Factor a perfect square on the left side: (d + 0.5)(d + 0.5) = 1 Calculate the square root of the right side: 1 Break this problem into two subproblems by setting (d + 0.5) equal to 1 and -1.Subproblem 1
d + 0.5 = 1 Simplifying d + 0.5 = 1 Reorder the terms: 0.5 + d = 1 Solving 0.5 + d = 1 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 = 1 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + d = 1 + -0.5 d = 1 + -0.5 Combine like terms: 1 + -0.5 = 0.5 d = 0.5 Simplifying d = 0.5Subproblem 2
d + 0.5 = -1 Simplifying d + 0.5 = -1 Reorder the terms: 0.5 + d = -1 Solving 0.5 + d = -1 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 = -1 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + d = -1 + -0.5 d = -1 + -0.5 Combine like terms: -1 + -0.5 = -1.5 d = -1.5 Simplifying d = -1.5Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.5, -1.5}
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