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Simplifying (d + -3)(d + 5) = 12 Reorder the terms: (-3 + d)(d + 5) = 12 Reorder the terms: (-3 + d)(5 + d) = 12 Multiply (-3 + d) * (5 + d) (-3(5 + d) + d(5 + d)) = 12 ((5 * -3 + d * -3) + d(5 + d)) = 12 ((-15 + -3d) + d(5 + d)) = 12 (-15 + -3d + (5 * d + d * d)) = 12 (-15 + -3d + (5d + d2)) = 12 Combine like terms: -3d + 5d = 2d (-15 + 2d + d2) = 12 Solving -15 + 2d + d2 = 12 Solving for variable 'd'. Reorder the terms: -15 + -12 + 2d + d2 = 12 + -12 Combine like terms: -15 + -12 = -27 -27 + 2d + d2 = 12 + -12 Combine like terms: 12 + -12 = 0 -27 + 2d + d2 = 0 Begin completing the square. Move the constant term to the right: Add '27' to each side of the equation. -27 + 2d + 27 + d2 = 0 + 27 Reorder the terms: -27 + 27 + 2d + d2 = 0 + 27 Combine like terms: -27 + 27 = 0 0 + 2d + d2 = 0 + 27 2d + d2 = 0 + 27 Combine like terms: 0 + 27 = 27 2d + d2 = 27 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 = 27 + 1 Reorder the terms: 1 + 2d + d2 = 27 + 1 Combine like terms: 27 + 1 = 28 1 + 2d + d2 = 28 Factor a perfect square on the left side: (d + 1)(d + 1) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (d + 1) equal to 5.291502622 and -5.291502622.Subproblem 1
d + 1 = 5.291502622 Simplifying d + 1 = 5.291502622 Reorder the terms: 1 + d = 5.291502622 Solving 1 + d = 5.291502622 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 = 5.291502622 + -1 Combine like terms: 1 + -1 = 0 0 + d = 5.291502622 + -1 d = 5.291502622 + -1 Combine like terms: 5.291502622 + -1 = 4.291502622 d = 4.291502622 Simplifying d = 4.291502622Subproblem 2
d + 1 = -5.291502622 Simplifying d + 1 = -5.291502622 Reorder the terms: 1 + d = -5.291502622 Solving 1 + d = -5.291502622 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 = -5.291502622 + -1 Combine like terms: 1 + -1 = 0 0 + d = -5.291502622 + -1 d = -5.291502622 + -1 Combine like terms: -5.291502622 + -1 = -6.291502622 d = -6.291502622 Simplifying d = -6.291502622Solution
The solution to the problem is based on the solutions from the subproblems. d = {4.291502622, -6.291502622}
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