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Simplifying 55 = (9 + -1d) * 11d Reorder the terms for easier multiplication: 55 = 11d(9 + -1d) 55 = (9 * 11d + -1d * 11d) 55 = (99d + -11d2) Solving 55 = 99d + -11d2 Solving for variable 'd'. Reorder the terms: 55 + -99d + 11d2 = 99d + -99d + -11d2 + 11d2 Combine like terms: 99d + -99d = 0 55 + -99d + 11d2 = 0 + -11d2 + 11d2 55 + -99d + 11d2 = -11d2 + 11d2 Combine like terms: -11d2 + 11d2 = 0 55 + -99d + 11d2 = 0 Factor out the Greatest Common Factor (GCF), '11'. 11(5 + -9d + d2) = 0 Ignore the factor 11.Subproblem 1
Set the factor '(5 + -9d + d2)' equal to zero and attempt to solve: Simplifying 5 + -9d + d2 = 0 Solving 5 + -9d + d2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + -9d + -5 + d2 = 0 + -5 Reorder the terms: 5 + -5 + -9d + d2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -9d + d2 = 0 + -5 -9d + d2 = 0 + -5 Combine like terms: 0 + -5 = -5 -9d + d2 = -5 The d term is -9d. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9d + 20.25 + d2 = -5 + 20.25 Reorder the terms: 20.25 + -9d + d2 = -5 + 20.25 Combine like terms: -5 + 20.25 = 15.25 20.25 + -9d + d2 = 15.25 Factor a perfect square on the left side: (d + -4.5)(d + -4.5) = 15.25 Calculate the square root of the right side: 3.905124838 Break this problem into two subproblems by setting (d + -4.5) equal to 3.905124838 and -3.905124838.Subproblem 1
d + -4.5 = 3.905124838 Simplifying d + -4.5 = 3.905124838 Reorder the terms: -4.5 + d = 3.905124838 Solving -4.5 + d = 3.905124838 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + d = 3.905124838 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + d = 3.905124838 + 4.5 d = 3.905124838 + 4.5 Combine like terms: 3.905124838 + 4.5 = 8.405124838 d = 8.405124838 Simplifying d = 8.405124838Subproblem 2
d + -4.5 = -3.905124838 Simplifying d + -4.5 = -3.905124838 Reorder the terms: -4.5 + d = -3.905124838 Solving -4.5 + d = -3.905124838 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + d = -3.905124838 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + d = -3.905124838 + 4.5 d = -3.905124838 + 4.5 Combine like terms: -3.905124838 + 4.5 = 0.594875162 d = 0.594875162 Simplifying d = 0.594875162Solution
The solution to the problem is based on the solutions from the subproblems. d = {8.405124838, 0.594875162}Solution
d = {8.405124838, 0.594875162}
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