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Simplifying d4 + -8d2 + 8 = 0 Reorder the terms: 8 + -8d2 + d4 = 0 Solving 8 + -8d2 + d4 = 0 Solving for variable 'd'. Begin completing the square. Move the constant term to the right: Add '-8' to each side of the equation. 8 + -8d2 + -8 + d4 = 0 + -8 Reorder the terms: 8 + -8 + -8d2 + d4 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + -8d2 + d4 = 0 + -8 -8d2 + d4 = 0 + -8 Combine like terms: 0 + -8 = -8 -8d2 + d4 = -8 The d term is -8d2. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8d2 + 16 + d4 = -8 + 16 Reorder the terms: 16 + -8d2 + d4 = -8 + 16 Combine like terms: -8 + 16 = 8 16 + -8d2 + d4 = 8 Factor a perfect square on the left side: (d2 + -4)(d2 + -4) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (d2 + -4) equal to 2.828427125 and -2.828427125.Subproblem 1
d2 + -4 = 2.828427125 Simplifying d2 + -4 = 2.828427125 Reorder the terms: -4 + d2 = 2.828427125 Solving -4 + d2 = 2.828427125 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + d2 = 2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + d2 = 2.828427125 + 4 d2 = 2.828427125 + 4 Combine like terms: 2.828427125 + 4 = 6.828427125 d2 = 6.828427125 Simplifying d2 = 6.828427125 Take the square root of each side: d = {-2.61312593, 2.61312593}Subproblem 2
d2 + -4 = -2.828427125 Simplifying d2 + -4 = -2.828427125 Reorder the terms: -4 + d2 = -2.828427125 Solving -4 + d2 = -2.828427125 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + d2 = -2.828427125 + 4 Combine like terms: -4 + 4 = 0 0 + d2 = -2.828427125 + 4 d2 = -2.828427125 + 4 Combine like terms: -2.828427125 + 4 = 1.171572875 d2 = 1.171572875 Simplifying d2 = 1.171572875 Take the square root of each side: d = {-1.0823922, 1.0823922}Solution
The solution to the problem is based on the solutions from the subproblems. d = {-2.61312593, 2.61312593, -1.0823922, 1.0823922}
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