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Simplifying d4 + -8d2 + 6 = 0 Reorder the terms: 6 + -8d2 + d4 = 0 Solving 6 + -8d2 + d4 = 0 Solving for variable 'd'. Begin completing the square. Move the constant term to the right: Add '-6' to each side of the equation. 6 + -8d2 + -6 + d4 = 0 + -6 Reorder the terms: 6 + -6 + -8d2 + d4 = 0 + -6 Combine like terms: 6 + -6 = 0 0 + -8d2 + d4 = 0 + -6 -8d2 + d4 = 0 + -6 Combine like terms: 0 + -6 = -6 -8d2 + d4 = -6 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 = -6 + 16 Reorder the terms: 16 + -8d2 + d4 = -6 + 16 Combine like terms: -6 + 16 = 10 16 + -8d2 + d4 = 10 Factor a perfect square on the left side: (d2 + -4)(d2 + -4) = 10 Calculate the square root of the right side: 3.16227766 Break this problem into two subproblems by setting (d2 + -4) equal to 3.16227766 and -3.16227766.Subproblem 1
d2 + -4 = 3.16227766 Simplifying d2 + -4 = 3.16227766 Reorder the terms: -4 + d2 = 3.16227766 Solving -4 + d2 = 3.16227766 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 = 3.16227766 + 4 Combine like terms: -4 + 4 = 0 0 + d2 = 3.16227766 + 4 d2 = 3.16227766 + 4 Combine like terms: 3.16227766 + 4 = 7.16227766 d2 = 7.16227766 Simplifying d2 = 7.16227766 Take the square root of each side: d = {-2.676243199, 2.676243199}Subproblem 2
d2 + -4 = -3.16227766 Simplifying d2 + -4 = -3.16227766 Reorder the terms: -4 + d2 = -3.16227766 Solving -4 + d2 = -3.16227766 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 = -3.16227766 + 4 Combine like terms: -4 + 4 = 0 0 + d2 = -3.16227766 + 4 d2 = -3.16227766 + 4 Combine like terms: -3.16227766 + 4 = 0.83772234 d2 = 0.83772234 Simplifying d2 = 0.83772234 Take the square root of each side: d = {-0.91527173, 0.91527173}Solution
The solution to the problem is based on the solutions from the subproblems. d = {-2.676243199, 2.676243199, -0.91527173, 0.91527173}
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