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Simplifying 4d4 + 5 = 14d2 Reorder the terms: 5 + 4d4 = 14d2 Solving 5 + 4d4 = 14d2 Solving for variable 'd'. Reorder the terms: 5 + -14d2 + 4d4 = 14d2 + -14d2 Combine like terms: 14d2 + -14d2 = 0 5 + -14d2 + 4d4 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.25 + -3.5d2 + d4 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + -3.5d2 + -1.25 + d4 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + -3.5d2 + d4 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + -3.5d2 + d4 = 0 + -1.25 -3.5d2 + d4 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 -3.5d2 + d4 = -1.25 The d term is -3.5d2. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5d2 + 3.0625 + d4 = -1.25 + 3.0625 Reorder the terms: 3.0625 + -3.5d2 + d4 = -1.25 + 3.0625 Combine like terms: -1.25 + 3.0625 = 1.8125 3.0625 + -3.5d2 + d4 = 1.8125 Factor a perfect square on the left side: (d2 + -1.75)(d2 + -1.75) = 1.8125 Calculate the square root of the right side: 1.346291202 Break this problem into two subproblems by setting (d2 + -1.75) equal to 1.346291202 and -1.346291202.Subproblem 1
d2 + -1.75 = 1.346291202 Simplifying d2 + -1.75 = 1.346291202 Reorder the terms: -1.75 + d2 = 1.346291202 Solving -1.75 + d2 = 1.346291202 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + d2 = 1.346291202 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + d2 = 1.346291202 + 1.75 d2 = 1.346291202 + 1.75 Combine like terms: 1.346291202 + 1.75 = 3.096291202 d2 = 3.096291202 Simplifying d2 = 3.096291202 Take the square root of each side: d = {-1.759628143, 1.759628143}Subproblem 2
d2 + -1.75 = -1.346291202 Simplifying d2 + -1.75 = -1.346291202 Reorder the terms: -1.75 + d2 = -1.346291202 Solving -1.75 + d2 = -1.346291202 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + d2 = -1.346291202 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + d2 = -1.346291202 + 1.75 d2 = -1.346291202 + 1.75 Combine like terms: -1.346291202 + 1.75 = 0.403708798 d2 = 0.403708798 Simplifying d2 = 0.403708798 Take the square root of each side: d = {-0.635380829, 0.635380829}Solution
The solution to the problem is based on the solutions from the subproblems. d = {-1.759628143, 1.759628143, -0.635380829, 0.635380829}
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