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Simplifying 12 = 4d2 + 8d + 2 Reorder the terms: 12 = 2 + 8d + 4d2 Solving 12 = 2 + 8d + 4d2 Solving for variable 'd'. Combine like terms: 12 + -2 = 10 10 + -8d + -4d2 = 2 + 8d + 4d2 + -2 + -8d + -4d2 Reorder the terms: 10 + -8d + -4d2 = 2 + -2 + 8d + -8d + 4d2 + -4d2 Combine like terms: 2 + -2 = 0 10 + -8d + -4d2 = 0 + 8d + -8d + 4d2 + -4d2 10 + -8d + -4d2 = 8d + -8d + 4d2 + -4d2 Combine like terms: 8d + -8d = 0 10 + -8d + -4d2 = 0 + 4d2 + -4d2 10 + -8d + -4d2 = 4d2 + -4d2 Combine like terms: 4d2 + -4d2 = 0 10 + -8d + -4d2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(5 + -4d + -2d2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(5 + -4d + -2d2)' equal to zero and attempt to solve: Simplifying 5 + -4d + -2d2 = 0 Solving 5 + -4d + -2d2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + 2d + d2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 2d + 2.5 + d2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 2d + d2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 2d + d2 = 0 + 2.5 2d + d2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 2d + d2 = 2.5 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 = 2.5 + 1 Reorder the terms: 1 + 2d + d2 = 2.5 + 1 Combine like terms: 2.5 + 1 = 3.5 1 + 2d + d2 = 3.5 Factor a perfect square on the left side: (d + 1)(d + 1) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (d + 1) equal to 1.870828693 and -1.870828693.Subproblem 1
d + 1 = 1.870828693 Simplifying d + 1 = 1.870828693 Reorder the terms: 1 + d = 1.870828693 Solving 1 + d = 1.870828693 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 = 1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + d = 1.870828693 + -1 d = 1.870828693 + -1 Combine like terms: 1.870828693 + -1 = 0.870828693 d = 0.870828693 Simplifying d = 0.870828693Subproblem 2
d + 1 = -1.870828693 Simplifying d + 1 = -1.870828693 Reorder the terms: 1 + d = -1.870828693 Solving 1 + d = -1.870828693 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 = -1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + d = -1.870828693 + -1 d = -1.870828693 + -1 Combine like terms: -1.870828693 + -1 = -2.870828693 d = -2.870828693 Simplifying d = -2.870828693Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.870828693, -2.870828693}Solution
d = {0.870828693, -2.870828693}
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