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Simplifying 4d2 + 9d + -81 = 0 Reorder the terms: -81 + 9d + 4d2 = 0 Solving -81 + 9d + 4d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -20.25 + 2.25d + d2 = 0 Move the constant term to the right: Add '20.25' to each side of the equation. -20.25 + 2.25d + 20.25 + d2 = 0 + 20.25 Reorder the terms: -20.25 + 20.25 + 2.25d + d2 = 0 + 20.25 Combine like terms: -20.25 + 20.25 = 0.00 0.00 + 2.25d + d2 = 0 + 20.25 2.25d + d2 = 0 + 20.25 Combine like terms: 0 + 20.25 = 20.25 2.25d + d2 = 20.25 The d term is 2.25d. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25d + 1.265625 + d2 = 20.25 + 1.265625 Reorder the terms: 1.265625 + 2.25d + d2 = 20.25 + 1.265625 Combine like terms: 20.25 + 1.265625 = 21.515625 1.265625 + 2.25d + d2 = 21.515625 Factor a perfect square on the left side: (d + 1.125)(d + 1.125) = 21.515625 Calculate the square root of the right side: 4.638493829 Break this problem into two subproblems by setting (d + 1.125) equal to 4.638493829 and -4.638493829.Subproblem 1
d + 1.125 = 4.638493829 Simplifying d + 1.125 = 4.638493829 Reorder the terms: 1.125 + d = 4.638493829 Solving 1.125 + d = 4.638493829 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + d = 4.638493829 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + d = 4.638493829 + -1.125 d = 4.638493829 + -1.125 Combine like terms: 4.638493829 + -1.125 = 3.513493829 d = 3.513493829 Simplifying d = 3.513493829Subproblem 2
d + 1.125 = -4.638493829 Simplifying d + 1.125 = -4.638493829 Reorder the terms: 1.125 + d = -4.638493829 Solving 1.125 + d = -4.638493829 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + d = -4.638493829 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + d = -4.638493829 + -1.125 d = -4.638493829 + -1.125 Combine like terms: -4.638493829 + -1.125 = -5.763493829 d = -5.763493829 Simplifying d = -5.763493829Solution
The solution to the problem is based on the solutions from the subproblems. d = {3.513493829, -5.763493829}
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