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Simplifying d(4d + 1) = 20 Reorder the terms: d(1 + 4d) = 20 (1 * d + 4d * d) = 20 (1d + 4d2) = 20 Solving 1d + 4d2 = 20 Solving for variable 'd'. Reorder the terms: -20 + 1d + 4d2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + 1d + 4d2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5 + 0.25d + d2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 0.25d + 5 + d2 = 0 + 5 Reorder the terms: -5 + 5 + 0.25d + d2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 0.25d + d2 = 0 + 5 0.25d + d2 = 0 + 5 Combine like terms: 0 + 5 = 5 0.25d + d2 = 5 The d term is 0.25d. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25d + 0.015625 + d2 = 5 + 0.015625 Reorder the terms: 0.015625 + 0.25d + d2 = 5 + 0.015625 Combine like terms: 5 + 0.015625 = 5.015625 0.015625 + 0.25d + d2 = 5.015625 Factor a perfect square on the left side: (d + 0.125)(d + 0.125) = 5.015625 Calculate the square root of the right side: 2.239559108 Break this problem into two subproblems by setting (d + 0.125) equal to 2.239559108 and -2.239559108.Subproblem 1
d + 0.125 = 2.239559108 Simplifying d + 0.125 = 2.239559108 Reorder the terms: 0.125 + d = 2.239559108 Solving 0.125 + d = 2.239559108 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + d = 2.239559108 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + d = 2.239559108 + -0.125 d = 2.239559108 + -0.125 Combine like terms: 2.239559108 + -0.125 = 2.114559108 d = 2.114559108 Simplifying d = 2.114559108Subproblem 2
d + 0.125 = -2.239559108 Simplifying d + 0.125 = -2.239559108 Reorder the terms: 0.125 + d = -2.239559108 Solving 0.125 + d = -2.239559108 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + d = -2.239559108 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + d = -2.239559108 + -0.125 d = -2.239559108 + -0.125 Combine like terms: -2.239559108 + -0.125 = -2.364559108 d = -2.364559108 Simplifying d = -2.364559108Solution
The solution to the problem is based on the solutions from the subproblems. d = {2.114559108, -2.364559108}
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