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Simplifying 4d2 + 12d + -30 = 0 Reorder the terms: -30 + 12d + 4d2 = 0 Solving -30 + 12d + 4d2 = 0 Solving for variable 'd'. Factor out the Greatest Common Factor (GCF), '2'. 2(-15 + 6d + 2d2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-15 + 6d + 2d2)' equal to zero and attempt to solve: Simplifying -15 + 6d + 2d2 = 0 Solving -15 + 6d + 2d2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + 3d + d2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 3d + 7.5 + d2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 3d + d2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 3d + d2 = 0 + 7.5 3d + d2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 3d + d2 = 7.5 The d term is 3d. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3d + 2.25 + d2 = 7.5 + 2.25 Reorder the terms: 2.25 + 3d + d2 = 7.5 + 2.25 Combine like terms: 7.5 + 2.25 = 9.75 2.25 + 3d + d2 = 9.75 Factor a perfect square on the left side: (d + 1.5)(d + 1.5) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (d + 1.5) equal to 3.122498999 and -3.122498999.Subproblem 1
d + 1.5 = 3.122498999 Simplifying d + 1.5 = 3.122498999 Reorder the terms: 1.5 + d = 3.122498999 Solving 1.5 + d = 3.122498999 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + d = 3.122498999 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + d = 3.122498999 + -1.5 d = 3.122498999 + -1.5 Combine like terms: 3.122498999 + -1.5 = 1.622498999 d = 1.622498999 Simplifying d = 1.622498999Subproblem 2
d + 1.5 = -3.122498999 Simplifying d + 1.5 = -3.122498999 Reorder the terms: 1.5 + d = -3.122498999 Solving 1.5 + d = -3.122498999 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + d = -3.122498999 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + d = -3.122498999 + -1.5 d = -3.122498999 + -1.5 Combine like terms: -3.122498999 + -1.5 = -4.622498999 d = -4.622498999 Simplifying d = -4.622498999Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.622498999, -4.622498999}Solution
d = {1.622498999, -4.622498999}
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