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Simplifying (d + 2)(d + 2) = 8 Reorder the terms: (2 + d)(d + 2) = 8 Reorder the terms: (2 + d)(2 + d) = 8 Multiply (2 + d) * (2 + d) (2(2 + d) + d(2 + d)) = 8 ((2 * 2 + d * 2) + d(2 + d)) = 8 ((4 + 2d) + d(2 + d)) = 8 (4 + 2d + (2 * d + d * d)) = 8 (4 + 2d + (2d + d2)) = 8 Combine like terms: 2d + 2d = 4d (4 + 4d + d2) = 8 Solving 4 + 4d + d2 = 8 Solving for variable 'd'. Reorder the terms: 4 + -8 + 4d + d2 = 8 + -8 Combine like terms: 4 + -8 = -4 -4 + 4d + d2 = 8 + -8 Combine like terms: 8 + -8 = 0 -4 + 4d + d2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 4d + 4 + d2 = 0 + 4 Reorder the terms: -4 + 4 + 4d + d2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 4d + d2 = 0 + 4 4d + d2 = 0 + 4 Combine like terms: 0 + 4 = 4 4d + d2 = 4 The d term is 4d. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4d + 4 + d2 = 4 + 4 Reorder the terms: 4 + 4d + d2 = 4 + 4 Combine like terms: 4 + 4 = 8 4 + 4d + d2 = 8 Factor a perfect square on the left side: (d + 2)(d + 2) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (d + 2) equal to 2.828427125 and -2.828427125.Subproblem 1
d + 2 = 2.828427125 Simplifying d + 2 = 2.828427125 Reorder the terms: 2 + d = 2.828427125 Solving 2 + d = 2.828427125 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + d = 2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + d = 2.828427125 + -2 d = 2.828427125 + -2 Combine like terms: 2.828427125 + -2 = 0.828427125 d = 0.828427125 Simplifying d = 0.828427125Subproblem 2
d + 2 = -2.828427125 Simplifying d + 2 = -2.828427125 Reorder the terms: 2 + d = -2.828427125 Solving 2 + d = -2.828427125 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + d = -2.828427125 + -2 Combine like terms: 2 + -2 = 0 0 + d = -2.828427125 + -2 d = -2.828427125 + -2 Combine like terms: -2.828427125 + -2 = -4.828427125 d = -4.828427125 Simplifying d = -4.828427125Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.828427125, -4.828427125}
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