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Simplifying (3d2 + 8d + -6) + (4d + -9d2 + 15) = 0 Reorder the terms: (-6 + 8d + 3d2) + (4d + -9d2 + 15) = 0 Remove parenthesis around (-6 + 8d + 3d2) -6 + 8d + 3d2 + (4d + -9d2 + 15) = 0 Reorder the terms: -6 + 8d + 3d2 + (15 + 4d + -9d2) = 0 Remove parenthesis around (15 + 4d + -9d2) -6 + 8d + 3d2 + 15 + 4d + -9d2 = 0 Reorder the terms: -6 + 15 + 8d + 4d + 3d2 + -9d2 = 0 Combine like terms: -6 + 15 = 9 9 + 8d + 4d + 3d2 + -9d2 = 0 Combine like terms: 8d + 4d = 12d 9 + 12d + 3d2 + -9d2 = 0 Combine like terms: 3d2 + -9d2 = -6d2 9 + 12d + -6d2 = 0 Solving 9 + 12d + -6d2 = 0 Solving for variable 'd'. Factor out the Greatest Common Factor (GCF), '3'. 3(3 + 4d + -2d2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(3 + 4d + -2d2)' equal to zero and attempt to solve: Simplifying 3 + 4d + -2d2 = 0 Solving 3 + 4d + -2d2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1.5 + -2d + d2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -2d + 1.5 + d2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -2d + d2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -2d + d2 = 0 + 1.5 -2d + d2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -2d + d2 = 1.5 The d term is -2d. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2d + 1 + d2 = 1.5 + 1 Reorder the terms: 1 + -2d + d2 = 1.5 + 1 Combine like terms: 1.5 + 1 = 2.5 1 + -2d + d2 = 2.5 Factor a perfect square on the left side: (d + -1)(d + -1) = 2.5 Calculate the square root of the right side: 1.58113883 Break this problem into two subproblems by setting (d + -1) equal to 1.58113883 and -1.58113883.Subproblem 1
d + -1 = 1.58113883 Simplifying d + -1 = 1.58113883 Reorder the terms: -1 + d = 1.58113883 Solving -1 + d = 1.58113883 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + d = 1.58113883 + 1 Combine like terms: -1 + 1 = 0 0 + d = 1.58113883 + 1 d = 1.58113883 + 1 Combine like terms: 1.58113883 + 1 = 2.58113883 d = 2.58113883 Simplifying d = 2.58113883Subproblem 2
d + -1 = -1.58113883 Simplifying d + -1 = -1.58113883 Reorder the terms: -1 + d = -1.58113883 Solving -1 + d = -1.58113883 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + d = -1.58113883 + 1 Combine like terms: -1 + 1 = 0 0 + d = -1.58113883 + 1 d = -1.58113883 + 1 Combine like terms: -1.58113883 + 1 = -0.58113883 d = -0.58113883 Simplifying d = -0.58113883Solution
The solution to the problem is based on the solutions from the subproblems. d = {2.58113883, -0.58113883}Solution
d = {2.58113883, -0.58113883}
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