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Simplifying 3d(d + 5) = 23 Reorder the terms: 3d(5 + d) = 23 (5 * 3d + d * 3d) = 23 (15d + 3d2) = 23 Solving 15d + 3d2 = 23 Solving for variable 'd'. Reorder the terms: -23 + 15d + 3d2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + 15d + 3d2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -7.666666667 + 5d + d2 = 0 Move the constant term to the right: Add '7.666666667' to each side of the equation. -7.666666667 + 5d + 7.666666667 + d2 = 0 + 7.666666667 Reorder the terms: -7.666666667 + 7.666666667 + 5d + d2 = 0 + 7.666666667 Combine like terms: -7.666666667 + 7.666666667 = 0.000000000 0.000000000 + 5d + d2 = 0 + 7.666666667 5d + d2 = 0 + 7.666666667 Combine like terms: 0 + 7.666666667 = 7.666666667 5d + d2 = 7.666666667 The d term is 5d. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5d + 6.25 + d2 = 7.666666667 + 6.25 Reorder the terms: 6.25 + 5d + d2 = 7.666666667 + 6.25 Combine like terms: 7.666666667 + 6.25 = 13.916666667 6.25 + 5d + d2 = 13.916666667 Factor a perfect square on the left side: (d + 2.5)(d + 2.5) = 13.916666667 Calculate the square root of the right side: 3.730504881 Break this problem into two subproblems by setting (d + 2.5) equal to 3.730504881 and -3.730504881.Subproblem 1
d + 2.5 = 3.730504881 Simplifying d + 2.5 = 3.730504881 Reorder the terms: 2.5 + d = 3.730504881 Solving 2.5 + d = 3.730504881 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + d = 3.730504881 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + d = 3.730504881 + -2.5 d = 3.730504881 + -2.5 Combine like terms: 3.730504881 + -2.5 = 1.230504881 d = 1.230504881 Simplifying d = 1.230504881Subproblem 2
d + 2.5 = -3.730504881 Simplifying d + 2.5 = -3.730504881 Reorder the terms: 2.5 + d = -3.730504881 Solving 2.5 + d = -3.730504881 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + d = -3.730504881 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + d = -3.730504881 + -2.5 d = -3.730504881 + -2.5 Combine like terms: -3.730504881 + -2.5 = -6.230504881 d = -6.230504881 Simplifying d = -6.230504881Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.230504881, -6.230504881}
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