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Simplifying d(8d + 1) = 40 Reorder the terms: d(1 + 8d) = 40 (1 * d + 8d * d) = 40 (1d + 8d2) = 40 Solving 1d + 8d2 = 40 Solving for variable 'd'. Reorder the terms: -40 + 1d + 8d2 = 40 + -40 Combine like terms: 40 + -40 = 0 -40 + 1d + 8d2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -5 + 0.125d + d2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 0.125d + 5 + d2 = 0 + 5 Reorder the terms: -5 + 5 + 0.125d + d2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 0.125d + d2 = 0 + 5 0.125d + d2 = 0 + 5 Combine like terms: 0 + 5 = 5 0.125d + d2 = 5 The d term is 0.125d. Take half its coefficient (0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. 0.125d + 0.00390625 + d2 = 5 + 0.00390625 Reorder the terms: 0.00390625 + 0.125d + d2 = 5 + 0.00390625 Combine like terms: 5 + 0.00390625 = 5.00390625 0.00390625 + 0.125d + d2 = 5.00390625 Factor a perfect square on the left side: (d + 0.0625)(d + 0.0625) = 5.00390625 Calculate the square root of the right side: 2.236941271 Break this problem into two subproblems by setting (d + 0.0625) equal to 2.236941271 and -2.236941271.Subproblem 1
d + 0.0625 = 2.236941271 Simplifying d + 0.0625 = 2.236941271 Reorder the terms: 0.0625 + d = 2.236941271 Solving 0.0625 + d = 2.236941271 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + d = 2.236941271 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + d = 2.236941271 + -0.0625 d = 2.236941271 + -0.0625 Combine like terms: 2.236941271 + -0.0625 = 2.174441271 d = 2.174441271 Simplifying d = 2.174441271Subproblem 2
d + 0.0625 = -2.236941271 Simplifying d + 0.0625 = -2.236941271 Reorder the terms: 0.0625 + d = -2.236941271 Solving 0.0625 + d = -2.236941271 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + d = -2.236941271 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + d = -2.236941271 + -0.0625 d = -2.236941271 + -0.0625 Combine like terms: -2.236941271 + -0.0625 = -2.299441271 d = -2.299441271 Simplifying d = -2.299441271Solution
The solution to the problem is based on the solutions from the subproblems. d = {2.174441271, -2.299441271}
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