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Simplifying 9(d + 5) = 3(d + -3) * 6d Reorder the terms: 9(5 + d) = 3(d + -3) * 6d (5 * 9 + d * 9) = 3(d + -3) * 6d (45 + 9d) = 3(d + -3) * 6d Reorder the terms: 45 + 9d = 3(-3 + d) * 6d Reorder the terms for easier multiplication: 45 + 9d = 3 * 6d(-3 + d) Multiply 3 * 6 45 + 9d = 18d(-3 + d) 45 + 9d = (-3 * 18d + d * 18d) 45 + 9d = (-54d + 18d2) Solving 45 + 9d = -54d + 18d2 Solving for variable 'd'. Combine like terms: 9d + 54d = 63d 45 + 63d + -18d2 = -54d + 18d2 + 54d + -18d2 Reorder the terms: 45 + 63d + -18d2 = -54d + 54d + 18d2 + -18d2 Combine like terms: -54d + 54d = 0 45 + 63d + -18d2 = 0 + 18d2 + -18d2 45 + 63d + -18d2 = 18d2 + -18d2 Combine like terms: 18d2 + -18d2 = 0 45 + 63d + -18d2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(5 + 7d + -2d2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(5 + 7d + -2d2)' equal to zero and attempt to solve: Simplifying 5 + 7d + -2d2 = 0 Solving 5 + 7d + -2d2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + -3.5d + d2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -3.5d + 2.5 + d2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -3.5d + d2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -3.5d + d2 = 0 + 2.5 -3.5d + d2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -3.5d + d2 = 2.5 The d term is -3.5d. 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. -3.5d + 3.0625 + d2 = 2.5 + 3.0625 Reorder the terms: 3.0625 + -3.5d + d2 = 2.5 + 3.0625 Combine like terms: 2.5 + 3.0625 = 5.5625 3.0625 + -3.5d + d2 = 5.5625 Factor a perfect square on the left side: (d + -1.75)(d + -1.75) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (d + -1.75) equal to 2.358495283 and -2.358495283.Subproblem 1
d + -1.75 = 2.358495283 Simplifying d + -1.75 = 2.358495283 Reorder the terms: -1.75 + d = 2.358495283 Solving -1.75 + d = 2.358495283 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 + d = 2.358495283 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + d = 2.358495283 + 1.75 d = 2.358495283 + 1.75 Combine like terms: 2.358495283 + 1.75 = 4.108495283 d = 4.108495283 Simplifying d = 4.108495283Subproblem 2
d + -1.75 = -2.358495283 Simplifying d + -1.75 = -2.358495283 Reorder the terms: -1.75 + d = -2.358495283 Solving -1.75 + d = -2.358495283 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 + d = -2.358495283 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + d = -2.358495283 + 1.75 d = -2.358495283 + 1.75 Combine like terms: -2.358495283 + 1.75 = -0.608495283 d = -0.608495283 Simplifying d = -0.608495283Solution
The solution to the problem is based on the solutions from the subproblems. d = {4.108495283, -0.608495283}Solution
d = {4.108495283, -0.608495283}
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